Contents
Introduction
iii
Chapter 1. Some fundamental concepts
Simple, semisimple and absolutely simple modules
Induction and restriction of modules
Clifford’s Theorem
Tensor products and tensor induction of modules
Representations and characters
Projective representations
1
1
5
10
11
14
19
Chapter 2. On tensor factorization
Factorization of representations
Factorization of characters
25
26
34
Chapter 3. On tensor induction
Two conjectures
Tensor induction and form induction
Linking the two conjectures
Examples
37
37
40
45
47
Chapter 4. Modules and bilinear forms
A further reduction
Bilinear forms on a simple module
Equivalence of forms
Induced forms
51
51
53
61
63
Chapter 5. Positive results
Induction from maximal subgroups
Some positive answers to the conjectures
A final remark
69
69
84
86
Chapter 6.
89
An example
i
ii
Appendix.
References
CONTENTS
Counting anisotropic submodules
95
109
Introduction
I. Representation Theory of finite groups may be recognized as the general research field of this thesis. The basic issue of this broad and fascinating subject is to consider (finite) groups, given in an abstract fashion,
and to study how they can be ‘realized’ as groups of matrices, whose structure is better understood and which are, in some sense, ‘concrete’. To be
more explicit, consider a group G, a field F, and a positive integer n; an
F-representation of degree n is a homomorphism from G to the general
linear group GL(n, F) (this is the group of all nonsingular n × n matrices
over F, endowed with the usual multiplication of matrices). It is clear that,
if V is an n-dimensional vector space over F, such a homomorphism provides a way of identifying G (or, more precisely, a quotient of G) with a
group of automorphisms of V , and, in this situation, we say that V acquires
a structure of FG-module.
All the problems discussed throughout this work concern (more or less
directly) a specific context in Representation Theory, which is the analysis of quasi-primitive representations. As it will be explained in Chapter 1, focusing on these objects is motivated by the following fact. If F
is an algebraically closed field of characteristic zero1, then the structure of
any F-representation for a finite group is well understood, provided quasiprimitive representations are under control 2. The process which leads to
this conclusion involves standard methods in Representation Theory, and it
will be described in detail in Chapter 1; at any rate, outlining it here will
give us the opportunity to explain what we mean by the additive structure
of a representation.
1A classical branch of Representation Theory deals with the complex field, which is
certainly a well distinguished member in the class of algebraically closed fields having
characteristic zero. Our discussion will start (in Chapter 2) with this setting, although it
will soon lead us to consider modules over finite fields.
2As we shall see, such a reduction is particularly effective for representations of finite
solvable groups.
iii
iv
INTRODUCTION
First of all, an F-representation D for a group G is called irreducible if,
considering an FG-module V associated to it, V is a simple FG-module.
This means that V does not have any (proper, nonzero) FG-submodule
(an FG-submodule of V is, by definition, a subspace which ‘admits’ the
action of the group of matrices D(G); in other words, it is setwise stabilized
by D(G)). The concept of irreducible representation is always crucial in
Representation Theory, but it is particularly important if the field F has
characteristic zero. In that case, all the F-representations of G can be
decomposed into the ‘direct sum’ of irreducible F-representations; therefore,
at this level, irreducible representations behave as ‘fundamental building
blocks’, and we can safely restrict our attention to them. Next, it is possible
to simplify our analysis further through the method of (ordinary) induction.
Without going here into many details, we just say that a good understanding
of an irreducible representation D for G is achieved, provided we can find
(and describe) a subgroup H , and a representation T for it, such that
D is induced by T from H . An effective method for recognizing such a
pair is yielded by a well known theorem of Clifford, iterated applications of
which enable us to find a pair (H, T ) such that D is induced by T from
H , and T is a quasi-primitive representation of H . We do not give here
the definition of quasi-primitive representation, but it is worth remarking
that a representation of this kind is irreducible, and the theorem of Clifford
mentioned above is not capable of giving any useful information on it; we can
therefore conclude that quasi-primitive representations are the fundamental
building blocks in this sense. The phenomena described in this paragraph
will be often alluded to as the additive structure of representations3.
II. The previous discussion was meant to suggest that a crucial step
in the analysis of representations is understanding the structure of quasiprimitive representations. In view of that, we can illustrate the line of
research followed in this work, starting from Chapter 2. Assume that D is a
quasi-primitive complex representation for a group G. Such a representation
can still be induced from a proper subgroup of G (but an important result
due to T. R. Berger excludes this possibility if G is solvable; see 1.21).
At any rate, we do not have any general method to exploit further the
3Of course it is not clear, from this brief sketch, in which sense ordinary induction
is an ‘additive process’. This fact is better understood by looking at induction from the
point of view of modules, and in Chapter 1 we shall actually follow this line. Just to
‘visualize’ the issue, the reader might want to look at Proposition 1.18(b).
INTRODUCTION
v
additive structure of D ; at this stage, it is natural to investigate D from
the point of view of its multiplicative structure, and here the concept of
tensor factorization comes across at first. More precisely, we are interested
in understanding the possible ways of factorizing D as an inner tensor product of projective representations4 of smaller degree. The literature contains
several results (see for example [FT]) which concern this topic, but they
serve different purposes (for instance, they seek a concept of ‘uniqueness’ for
factorizations) and do not aim to provide all factorizations; indeed, in some
cases (even when the group is solvable) there are factorizations but they do
not provide any. In Chapter 2 we consider a different aspect of the problem:
rather than insisting on uniqueness of factorizations, our aim is to control
and parametrize, in terms of the group structure of G, all the possible ways
of decomposing a quasi-primitive representation D (or a quasi-primitive character χ) in the product of two factors. In particular, Corollary 2.10, and its
translation in the language of characters, Theorem 2.12, show that there is
an explicit bijection between the set of all the two-factors decompositions5 of
D (of χ) and a particular interval in the lattice of normal subgroups of G,
provided a natural extra assumption on D (on χ) is made. This result
follows from the more general Theorem 2.8.
III. The next step, discussed in the first part of Chapter 3, is to consider a situation in which tensor factorization does not help to simplify the
structure of the given representation D . More explicitly, we assume that
D is a faithful (see 1.26(a)), quasi-primitive, and tensor-indecomposable representation of G. To begin with, the existence of such a representation
implies a strong restriction on the structure of G. Namely, denoting by F
the Fitting subgroup (which is assumed noncentral) of G, and by Z the
centre of G, the quotient group F/Z turns out to be itself a simple module
for G over a prime field. Moreover, the module F/Z carries a nonsingular
symplectic form which is G-invariant; in other words, G acts on F/Z as
a group of isometries with respect to a particular nonsingular symplectic
form. This is the natural context in which the method of tensor induction
4The concept of projective representation, which generalizes the usual one, has to be
introduced in such a multiplicative setting; the reason for this will be clarified in the last
section of Chapter 1.
5The scope of the word ‘explicit’, as well as the rigorous meaning of ‘two-factors
decomposition’ (what is factorized is indeed the projective-equivalence class of D ) will be
clear from the discussion of Chapter 2.
vi
INTRODUCTION
plays a significant role (tensor induction is a process defined in strong analogy with ordinary induction, and it can be thought as a transposition of
it to a multiplicative context). In particular, we can ask whether the representation D is tensor-induced by a projective representation of a proper
subgroup H of G. The deep link between tensor induction for D and the
additive structure of the symplectic module F/Z is observed and discussed
by T. R. Berger in [Be] and in [Be2]. Moreover, a theorem by L. G. Kovács
([Kov], Section 6) can be paraphrased by saying that D is tensor-induced
by a projective representation of H if F/Z is form-induced from H (form
induction is a kind of ordinary induction in which also the structure given by
the symplectic form is taken in account; see Definition 3.9). In Theorem 3.13
we complete the picture by proving that, if D is a faithful, quasi-primitive,
tensor-indecomposable representation of G, and if H is a subgroup of G,
then there exists an explicit bijection between the set of (equivalence classes
of ) projective representations of H which tensor-induce D , and the set of
H -submodules of F/Z which form-induce F/Z .
IV. Through the previous discussion, we outlined a significant analogy
between additive and multiplicative methods in Representation Theory of finite groups: tensor factorization ‘corresponds’ to direct sum decomposition,
whereas tensor induction corresponds to ordinary induction. Nevertheless,
as we shall see in many occasions, such an analogy turns out to be not at all
complete. In the last part of this introduction we present the main problem
discussed in the thesis; the analysis of it will emphasize several aspects of
this incompleteness.
As it is well known, the concept of ordinary induction is closely related
to the concept of restriction (indeed, this fact can be meaningfully expressed
in the language of categories). One of the features of such a good mutual behaviour is the following: let D be an irreducible representation of G, and T
a representation of the subgroup H ; then D is induced by T from H if and
only if T appears as a direct summand in D↓H and deg D = |G : H| deg T
(here D↓H is the restriction of D to the subgroup H , which is obviously
a representation for H , whereas |G : H| denotes the index of H in G).
Since tensor induction can be thought as the multiplicative counterpart of
induction, one could try to formulate a parallel statement as follows.
Let D be a faithful, quasi-primitive and tensor-indecomposable representation of G, and P a projective representation of the subgroup H . Then D
INTRODUCTION
vii
is tensor-induced by P from H if and only if P appears as a tensor factor
in D↓H and deg D = (deg P )|G:H| .
In a weaker form, one may ask whether (in the same setting)
D is tensor-induced from H if and only if D ↓H has a tensor factor
of the ‘right’ degree (D not being tensor-induced necessarily by that tensor
factor).
The core of our work consists in carrying out an analysis of this problem,
in the context of solvable groups (as we mentioned, that is the context in
which this kind of concepts and methods turns out to be particularly important). The statement above, whose formulation is due to L. G. Kovács,
appears in the thesis as Conjecture 3.2 (which has a strong and a weak version), and the original motivation of it is to seek a way of characterizing
tensor induction in an ‘internal’ fashion (ordinary induction does have such
an internal characterization). As it will be explained (see, for instance, the
footnote on page 39), even the weak version of Conjecture 3.2 would provide,
if confirmed, a good test for tensor induction, since the results of Chapter 2
yield a control of all the possible factorizations of D↓H . We also mention
that, very recently, some effective computational algorithms for the internal
recognition of tensor products (and of tensor induction) have been developed
and implemented by C. R. Leedham-Green and E. A. O’Brien in [LGO].
Our approach to Conjecture 3.2 consists of a chain of subsequent reductions, which may be outlined as follows. First of all, the characterization
of tensor induction achieved in Chapter 3 enables us to change our point
of view, transposing the problem into a more ‘comfortable’ additive setting.
Indeed, Conjecture 3.2 appears to be deeply linked to a statement (Conjecture 3.3) which establishes a connection between (ordinary) induction and
form induction for symplectic modules. Conjecture 3.3 presents a strong
and a weak formulation as well.
At this first level of the mentioned chain of reductions, it is already possible to show that Conjectures 3.2 and 3.3 both fail in their strong version,
and also in the weak one if the index of the subgroup H in G is not assumed
to be odd (this is achieved via Examples 3.17 and 3.18, and yields to the
conclusion that we can not pursue the analogy between ordinary and tensor
induction as far as we could initially hope). What is then left is to concentrate on the weak version of the conjectures, with the additional assumption
that |G : H| is odd.
viii
INTRODUCTION
The subsequent reduction, expressed by Conjecture 4.1, is obtained by
means of a deeper analysis of the relationship between modules and bilinear
forms, which is developed in Chapter 4. Given that, we are in a position
to obtain (in Chapter 5) some positive results. Namely, the weak forms
of Conjectures 3.2 and 3.3 are proved to be true for normal subgroups of
odd index (see Theorems 5.23 and 5.24), and this, together with Example
3.18, provides a full understanding of what happens in this context with
respect to normal subgroups. As regards subgroups which are not necessarily
normal, Theorems 5.20 and 5.21 show that the two (weak) conjectures are
true for subgroups whose index is an (odd) prime. These are proved after
two crucial results (Theorems 5.14 and 5.18), concerning the structure of
modules induced from maximal subgroups, are established. At this stage
it is also worth remarking that Example 3.17, which disproves the strong
versions of 3.2 and 3.3, involves a normal subgroup of odd prime index.
What remains to be understood is whether Conjectures 3.2 and 3.3 are
valid for not normal subgroups whose index is odd, but not necessarily a
prime. This is left as an open problem. However, the discussion of Chapter 5
suggests that extending 5.20 and 5.21 to subgroups having index a power of
an (odd) prime could be sufficient to achieve the final answer.
Chapter 6, which ends the main matter of the thesis, essentially consists
of an example (Example 6.2). The aim of it is to clarify several issues arising
from Chapters 4 and 5, and to prove that certain assumptions which appear
in statements from those chapters are really needed. Moreover, Example 6.2
appears as a ‘summary’ of awkward behaviours of form induction (with
respect to ordinary induction), and it outlines unexpected situations in the
structure of induced modules.
Finally, the appendix of the thesis reports an earlier approach to our
main problem, which is capable of proving that (weak) Conjectures 3.2
and 3.3 are true for normal subgroups of odd index; here we follow a completely different line, which involves an explicit computation of the number
of anisotropic simple submodules in a symplectic module.
This concludes our introductory overview; a more precise and exhaustive description of the material discussed in the thesis can be found in the
preambles (or, when there is not a preamble, in the initial sections) of the
various chapters.
CHAPTER 1
Some fundamental concepts
The main purpose of this preliminary chapter is to provide an overview
of concepts and methods in Representation Theory, starting from a very
general context, and eventually focusing on the more specific environment
of our discussion. At the same time, a good deal of notation and conventions
which will be relevant in the sequel are introduced.
Coming to this point, throughout the whole thesis all the groups generically denoted by G (or H ) are meant to be finite, and all the vector spaces
will have finite dimension.
Also, we tend to use the right notation for maps; in particular, if a
vector is given in coordinates, the action of a matrix on it is always obtained
regarding the vector as a row, and multiplying it on the right by the relevant
matrix. Nevertheless, we follow the well consolidated tradition of writing
representations and characters on the left; then, for the sake of coherence,
we adopt the ‘left convention’ for all the maps when representations or characters are involved in the discussion.
Simple, semisimple and absolutely simple modules
We start with some basic definitions.
1.1. Definition. Let G be a group, and F a field.
(a) Let V be a vector space over F; if a group homomorphism % from G to
AutF (V ) is given, then we say that V is an FG-module (by means of %).
In this situation every element of G ‘acts’ on V , the action being defined
(and denoted) by
v g := v(g%).
(b) Let V1 and V2 be FG-modules. If there exists an isomorphism of vector
spaces α : V1 → V2 such that (v g )α = (vα)g holds for all v in V1 and
g in G, then we say that V1 and V2 are isomorphic FG-modules. This
defines an equivalence relation on the set of FG-modules.
1
2
1. SOME FUNDAMENTAL CONCEPTS
(c) Let V be an FG-module, and W an F-subspace of V . If W ‘admits’
the action of G, in the sense that wg is an element of W for all w in W
and g in G, then W is called a submodule of V . If the only submodules
of V are the zero space and V itself, then we say that V is a simple
FG-module, or also an irreducible FG-module (but the zero space itself
is never taken in account as a simple module). Finally, if V has a direct
decomposition in which every direct summand is a simple submodule,
then we say that V is a semisimple FG-module.
Observe that any vector space V over F can be endowed with a ‘trivial’
structure of FG-module for any group G, considering the group homomorphism from G to AutF (V ) which maps every element of G to the identity.
If the dimension of V is n, then this module is called the n-dimensional
trivial FG-module.
It is worth mentioning, without going into details, that the string of
symbols ‘FG’ is not just meant to suggest that G is the relevant group and
F the relevant field, but in this context it denotes the group algebra of G
over F (see [Is], Chapter 1). Indeed, any group homomorphism from G to
AutF (V ) turns out to be the restriction of a uniquely determined algebra
homomorphism from FG to EndF (V ), and therefore the objects defined in
1.1(a) are acted upon by the whole group algebra. Also, the same notation
FG is used for the (right) regular module of G over F; this is the FG-module
defined as follows: regard the group algebra FG as a vector space, forgetting
the ring structure, and let G act on it by right multiplication. Usually no
confusion arises from having such an ambiguous notation, as the relevant
meaning of the symbols is suggested by the context.
We also remark that simplicity and semisimplicity have a good behaviour
with respect to the concept of isomorphism introduced in 1.1(b), in the sense
that every module which is isomorphic to a simple (semisimple) one is also
simple (semisimple).
We shall spend some more words on semisimple modules; first of all, a
well known theorem of Maschke provides a full characterization of the condition in which all the modules for a group over a given field are semisimple.
1.2. Theorem. Let G be a group, and F a field. Then the following conditions are equivalent:
(a) all the FG-modules are semisimple,
(b) the characteristic of F does not divide the order of G.
SIMPLE, SEMISIMPLE AND ABSOLUTELY SIMPLE MODULES
3
Next, consider a semsimple FG-module V ; by definition, V has a direct
decomposition in which the summands are simple submodules, say
V = V11 ⊕ V21 ⊕ · · · ⊕ Vk11 ⊕ V12 ⊕ · · · ⊕ Vk22 ⊕ · · · ⊕ V1t ⊕ · · · ⊕ Vktt .
Here the indices are meant to suggest that two summands Vsr and Vnm are
isomorphic if and only if r is equal to m. The direct summands in such
a decomposition are uniquely determined only up to isomorphism, and are
called the simple constituents of V ; but if we define the submodule Wi of
V by Wi := V1i ⊕ . . . ⊕ Vkii (where i is in {1, . . . , t}), then we get
V = W1 ⊕ · · · ⊕ Wt
and this decomposition is totally unique. Moreover, each of the Wi has
a unique direct complement in V , namely the sum of the others. The
submodules Wi are called the homogeneous components of V , and each
of those is precisely the sum of all the simple submodules of V belonging
to a particular isomorphism type. In general, a semisimple module which
consists of a unique homogeneous component is called homogeneous.
Let now G be a group, F a field, and V1 , V2 FG-modules; we denote by
HomFG (V1 , V2 ) the F-subspace of HomF (V1 , V2 ) defined as follows: if ε is an
element of HomF (V1 , V2 ), then ε is in HomFG (V1 , V2 ) if (v g )ε = (vε)g holds
for all v in V and g in G (such an ε is called an FG-homomorphism from
V1 to V2 ). We introduce here this terminology because in the statement
of Schur’s Lemma, which is presented below as Lemma 1.3, we meet the
endomorphism ring EndFG (V ) of the FG-module V . This set is actually a
subalgebra of EndF (V ), and it contains always a copy of F acting on V by
scalar multiplication.
1.3. Lemma. Let G be a group, F a field, and V a simple FG-module.
Then EndFG (V ) is a division ring. Moreover, if F is algebraically closed,
then we have EndFG (V ) = F.
As an immediate consequence of Schur’s Lemma (and of a well known
theorem of Wedderburn, which asserts that any finite division ring is a field)
we see that, if F is a finite field, then EndFG (V ) is a (finite) field as well.
We conclude the section introducing some more concepts which will come
across in Chapter 5.
1.4. Definition. Let G be a group, F a field, and V an FG-module. If V
is simple and EndFG (V ) = F, then we say that V is absolutely simple (or
absolutely irreducible).
4
1. SOME FUNDAMENTAL CONCEPTS
1.5. Definition. Let G be a group, F a field, and V an FG-module. If
K is a field extension of F having finite degree, then we denote by V K the
KG-module whose underlying set is V ⊗F K, the actions of K and G being
defined as follows: let {v1 , . . . vn } and {ε1 , . . . εm } be bases over F for V
and K respectively (here of course K is viewed as a vector space over F);
we set
(vi ⊗ εj )k := vi ⊗ εj k
for all i and j in the relevant sets of indices, and for all k in K. Then
these relations can be extended by F-linearity to the whole V ⊗F K, as
{vi ⊗ εj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} is a basis for V ⊗F K over F, and in this
way a structure of K-vector space arises on V ⊗F K. Now, {vi ⊗1 : 1 ≤ i ≤ n}
is a K-basis for V ⊗F K, and we can define an action of G on this space
extending by K-linearity the relations
(vi ⊗ 1)g := vig ⊗ 1
for all i in {1, . . . , n} and g in G.
We remark that Definition 1.5 can be given, in full generality, for any
field extension of F (see [HB], VII, 11.1); here we sketched the construction
in the finite degree case for the sake of simplicity, and because after all we
shall be dealing only with that situation.
The connection between the two definitions above lies in the following
1.6. Proposition. Let G be a group, F a field, and V a simple FG-module.
Then V is absolutely simple if and only if, for any field extension K of F,
the KG-module V K is also simple.
(See [HB], VII, 2.2).
As regards absolute simplicity, we also have to mention the concept of
splitting field ; after the definition, we recall the statement of a theorem
which will turn out to be useful in our discussion.
1.7. Definition. Let G be a group, and F a field; we say that F is a
splitting field for G if all the simple FG-modules are absolutely simple.
1.8. Theorem. Let G be a group, and F a field. There exists a finite degree
extension K of F such that K is a splitting field for all the subgroups of G
(including G itself ).
(See [HB], VII, 2.6).
INDUCTION AND RESTRICTION OF MODULES
5
Induction and restriction of modules
Following the discussion in the previous section it is clear that, if we are
dealing with semisimple modules (for example, if we are looking at a situation in which Maschke’s Theorem applies), then we can reasonably restrict
our attention to the analysis of simple modules, which are the ‘fundamental
building blocks’ in this context.
In general the additive structure of a simple module can be very rich,
in spite of simplicity, and it is possible to exploit this additive complexity
through the method of induction of modules, achieving crucial informations
about the structure of the module itself. We shall introduce this particular
application of induction (which is very relevant from our point of view)
in the last part of the section, drawing some conclusions in the following
one; but, first of all, we present here the concept of induction in its original
fashion (that is, as a standard method in Representation Theory for building
a ‘larger’ module from a given one), and also some aspects of the relationship
between induction and restriction of modules.
There are several ways, of course all leading to the same final result, for
defining the process of induction; the one we sketch here is probably not
the best available from a certain point of view, but it will turn out to be
convenient for our purposes (see Remark 1.10).
1.9. Definition. Let G be a group, H a subgroup of G whose index is n,
F a field, and W an FH -module. We construct an FG-module, denoted by
W↑G , and such that dim W↑G = n dim W , following two steps.
(a) Since we are constructing a module whose dimension is meant to be
n dim W , a convenient underlying vector space is given by the external
direct sum of n copies of W ; we shall denote this space by W ⊕n (the
elements of such a space are ordered n-tuples of elements in W ). It is
clear that the group H n , defined as the direct product of n copies of
H , can act ‘componentwise’ on W ⊕n , that is
h
(w1 , . . . , wn )(h1 ,...,hn ) := (w1 1 , . . . , wnhn ).
Also, the symmetric group Sn of all permutations of {1, . . . , n} acts
both on H n and W ⊕n by ‘permuting coordinates’: for any ξ in Sn , we
set
(h1 , . . . , hn )ξ := (h1ξ−1 , . . . , hnξ−1 ),
6
1. SOME FUNDAMENTAL CONCEPTS
and
(w1 , . . . , wn )ξ := (w1ξ−1 , . . . , wnξ−1 ).
The semidirect product H n oSn formed according to this action is (by
definition) the wreath product H o Sn , and it can be checked that the
actions of H n and Sn on W ⊕n ‘match up’ conveniently, so that W ⊕n
acquires a structure of F[H o Sn ]-module.
(b) Let Ω be the set of right H -cosets in G, and let π be the permutation
representation of G on Ω given by right multiplication; also, let us
fix a right transversal {g1 , . . . , gn } for H in G (recall that the subset
{g1 , . . . , gn } of G is a right transversal for H if G is the disjoint union
of the right H -cosets Hg1 , . . . , Hgn ). Now, given an element x in G,
we have
gi x = h(i, x)gi(xπ)
for all i in {1, . . . , n}, where h(i, x) is an element of H uniquely determined by i and x. Next, consider the map ϕ : G → H o Sn defined
by
xϕ := (h(1, x), . . . , h(n, x))(xπ).
It can be shown (see [CR], 13.3) that ϕ is a monomorphism of groups,
which we call, following [Be], a Frobenius embedding; therefore G is
embedded in H o Sn , and the desired structure of FG-module on W ⊕n
is now defined by restriction (see 1.12). We denote this module by W↑G
(but sometimes, to avoid ambiguity or for the sake of emphasis, we use
G
the notation W ↑G
H ), and any FG-module which is isomorphic to W ↑
is said to be induced by W from H .
1.10. Remark. It is important to observe that the construction above does
not provide a good definition for the module W ↑G (strictly speaking, we
should not even say ‘the’ module W↑G ). Indeed, several choices are involved
in the process (for example, a right transversal for H in G has to be fixed),
and making different choices yields different modules. But the isomorphism
type of W↑G is uniquely determined, so that the previous definition is safe;
moreover, if Z is an FH -module isomorphic to W , then Z↑G is isomorphic
to W↑G as well.
It is possible to give a completely choice-free definition of the FG-module
induced by the FH -module W , as the tensor product, over the algebra FH ,
of the regular module FG with W (we are not going into details here; see
[HB], VII, 4.1). In this sense the way we chose for defining induction is
INDUCTION AND RESTRICTION OF MODULES
7
not the best one, but it has the advantage that it can be easily transposed
to a multiplicative context (see also the introductory discussion to Definition 1.25).
The following proposition points out a very useful feature of induction.
Roughly speaking, we see that the process of induction is ‘transitive’.
1.11. Proposition. Let G be a group, H and K subgroups of G such that
H is contained in K , F a field, and W an FH -module; then the FG-module
K G
W↑G
H is isomorphic to (W↑H )↑K .
We have seen how induction enables us to construct a module for the
group G, starting from a module for the subgroup H of G. There is a
very natural method to do the converse, that is, to ‘construct’ a module
for H starting from a module for G, and this is provided by the process of
restriction (which we already met in 1.9(b)). If V is an FG-module, then
V itself can be viewed as an FH -module simply forgetting the rest of the
group G. This is expressed in the following
1.12. Definition. Let G be a group, F a field, and V an FG-module. If %
is the group homomorphism from G to AutF (V ) which yields the relevant
structure of FG-module, then the restriction of % to H yields a structure of
FH -module on the underlying vector space of V . The FH -module which
arises in this way is denoted by V ↓H .
1.13. Remark. Consider the setting of Definition 1.9; if the transversal
used in constructing W↑G is indexed so that g1 lies in H , then the map
−1
W → (W↑G )↓H , w 7→ (wg1 , 0, . . . , 0)
is a homomorphism of FH -modules, which is indeed injective. Thus, we
can say that W is a submodule of (W↑G )↓H , tacitly identifying W with its
image along this embedding.
The concepts of induction and restriction of modules are very deeply
intertwined; this fact can be conveniently expressed and described in the
language of categories, saying that induction provides an exact covariant
functor from the category of FH -modules to the category of FG-modules
(here of course H is a subgroup of G), whereas restriction provides a covariant exact functor which goes the other way round; moreover, these two
are adjoint functors of each other. A very important feature of this good
8
1. SOME FUNDAMENTAL CONCEPTS
mutual behaviour is the so called Nakayama reciprocity, which is expressed
by the theorem below.
1.14. Theorem. Let G be a group, H a subgroup of G, V an FG-module,
and W an FH -module. Then HomFG (W ↑G , V ) is isomorphic, as an
F-vector space, to HomFH (W, V ↓H ), and HomFG (V, W ↑G ) is isomorphic
to HomFH (V ↓H , W ).
1.15. Remark. The first claim of 1.14 is implied by the following ‘universal property’ of induced modules (see [HB], VII, 4.4a)), which we spell
out here as often we shall take advantage of it. In the setting of 1.14,
given any FH -homomorphism α from W to V ↓H , there exists a unique
FG-homomorphism ᾱ from W↑G to V such that wᾱ = wα holds for all w
in W (this makes sense, adopting the convention established in 1.13).
It ought to be mentioned that the second claim of Theorem 1.14 comes
more properly from the context of coinduction (see [HB], VII, 4.6); but in a
finite setting, such as ours, we need not distinguish between induction and
coinduction.
Consider now an FH -module W ; not very much can be said on the
structure of the FH -module (W↑G )↓H (indeed, our main task in Chapter 5
will be to provide some understanding of this structure in a relevant situation), but very useful informations are yielded by the result presented below
as Lemma 1.17, which is known as Mackey’s Lemma. Before stating it, we
have to introduce some terminology.
1.16. Definition. Let G be a group, H a subgroup of G, F a field, and W
an FH -module; also, let g be an element of G. It is possible to define an
F[H g ]-module as follows: take an isomorphic copy W g := {wg : w ∈ W }
of the underlying vector space of W (so that w 7→ wg is a vector space isog
morphism), and define the action of H g on W g by setting (wg )h := (wh )g
for all w in W and h in H ; the F[H g ]-module which arises in this way is
denoted by W g , and it is called the translate of W by g . It is easily checked
that W g is simple if and only if W is so.
Observe that, If V is an FG-module and W is a submodule of V ↓H , then
its translates defined above are exactly what we expect. Indeed, the image
of W under the action of g on V has a natural structure of F[H g ]-module
(H g acts on it by restriction of the original action of G) and, with this
structure, it turns out to be F[H g ]-isomorphic to W g . We shall make no
INDUCTION AND RESTRICTION OF MODULES
9
distinction between such concrete translates and the translates abstractly
defined above.
Now we can present the statement of Mackey’s Lemma (see [CR], 10.13).
1.17. Lemma. Let G be a group, H and K subgroups of G, F a field, and
W an FH -module; also, let R be a subset of G such that G is the disjoint
union of the double cosets HrK , for r running in R. Then we have
M
(W↑G )↓K '
[(W r )↓H r ∩K ]↑K .
r∈R
In particular, if G = HK , then the formula above becomes
(W↑G )↓K ' (W↓H∩K )↑K .
In 1.23 we shall see that, if V is an FG-module, it is also possible to
have some control of the structure of (V ↓H )↑G .
So far we presented induction as a method for building new modules, but
since it is clear that the structure of an induced module W↑G is under a good
control (as soon as the structure of W is so), it is natural to look at induction
also from a different point of view. Indeed, given a simple FG-module V ,
the problem of studying the structure of V is very much simplified if we
can decide whether V is induced or not from a proper subgroup of G, even
better if we can actually recognize a subgroup H , together with a submodule
W of V ↓H , such that V is isomorphic to W↑G . We shall come back to this
point in the next section; what is relevant here is to stress that induction
may provide a convenient ‘internal’ description of a module, and not only an
external construction. For this purpose we present below some very useful
properties of internal induction.
1.18. Proposition. Let G be a group, H a subgroup of G, F a field, V a
simple FG-module, and W an FH -module. Then the following properties
hold:
(a) V is induced by W from H if and only if dim V = |G : H| dim W and
V ↓H has a submodule isomorphic to W .
(b) If W is a submodule of V ↓H , then V is induced by W from H if and
only if, regarding V as a vector space, we have
M g
V =
W
g∈T
where T is any right transversal for H in G.
10
1. SOME FUNDAMENTAL CONCEPTS
Observe that, in the setting of the above proposition, V ↓H can not have
any submodule whose dimension is less than (dim V )/|G : H|. Of course
any submodule having precisely that dimension induces V from H , and
it is a direct summand of V ↓H . Moreover, it is not hard to see that any
submodule of V ↓H whose codimension is (dim V )/|G : H| is also a direct
summand of V ↓H .
Clifford’s Theorem
In this section we shall see how induction can be exploited to get useful
informations on the structure of simple modules; the other critical ingredient
which is needed for this purpose is a well known theorem of Clifford. Before
stating it (see [CR], 11.1 for a proof) we recall that, if G is a group, N a
normal subgroup of G, F a field, and X an FN -module, then the translate
X g (see Definition 1.16) is also an FN -module for all g in G, and we can
define the set IG (X) := {g ∈ G : X g is FN -isomorphic to X}. This set is
actually a subgroup of G, and it is called the inertia subgroup of X in G.
1.19. Theorem. Let G be a group, N a normal subgroup of G, F a field,
and V a simple FG-module. Then the following properties hold.
(a) V ↓N is semisimple, and therefore we have
V ↓N = V1 ⊕ · · · ⊕ Vl ,
where the Vi are the homogeneous components of V ↓N .
(b) The action of G on V yields a permutation action of G on the set
{V1 , . . . , Vl }; this permutation action is transitive, and its kernel contains N .
(c) All the simple submodules of V ↓N have the same dimension.
(d) If X is a simple submodule of V ↓N lying in V1 , then I := IG (X) is
precisely the stabilizer of V1 in the permutation action mentioned in (b).
Moreover, V1 is a simple FI -module, and we have V ' V1↑G
I .
Let us consider now a simple FG-module V ; if our aim is to recognize
a proper subgroup of G from which V is induced (if there is any), then
we can just check the restriction of V to normal subgroups. As soon as we
find a normal subgroup N of G such that V ↓N is inhomogeneous, then
the stabilizer I in G of a homogeneous component, say V1 , is a proper
subgroup of G and, by Theorem 1.19(d), we get V ' V1 ↑G
I . Now V1 is
a simple FI -module, so that we can iterate the previous step; the process
TENSOR PRODUCTS AND TENSOR INDUCTION OF MODULES
11
eventually stops when we reach a pair (H, W ), where H is a subgroup of
G and W is an FH -module, such that the restriction of W to any normal
subgroup of H is homogeneous (such a pair is called a stabilizer limit for
the pair (G, V )). Observe that, by transitivity of induction, V is induced
by W from H .
At this stage some further terminology is needed.
1.20. Definition. Let G be a group, F a field, and V a simple FG-module.
Then V is called primitive if it is not induced from any proper subgroup
of G, whereas it is called quasi-primitive if its restriction to any normal
subgroup of G is homogeneous.
By what we observed above it is totally straightforward, as the terminology suggests, that a primitive module is also quasi-primitive.
A natural question could be, conversely, how much of the additive structure is left in a quasi-primitive module; more explicitly, how ‘far’ a quasiprimitive module is from being primitive at all. Roughly speaking, the
process described above is likely to be effective, in spoiling the additive
structure of the given module, insofar as we have abundance of normal subgroups. Indeed, the concept of quasi-primitivity is completely useless for
simple groups, whereas it is very powerful in a solvable context (working
over an algebraically closed field), as the following theorem of Berger shows
(see [Is], 11.33). The statement of Berger’s theorem will conclude our preliminary overview of the additive stucture of modules.
1.21. Theorem. Let G be a solvable group, F an algebraically closed field,
and V a simple FG-module. Then V is primitive if and only if it is quasiprimitive.
Tensor products and tensor induction of modules
We need now to establish some terminology and notation which are
relevant in the analysis of the multiplicative structure of modules.
1.22. Definition. Let G be a group, F a field, and V1 , V2 FG-modules;
the inner tensor product of V1 and V2 is an FG-module, whose dimension
is (dim V1 )(dim V2 ), which is denoted by V1 ⊗ V2 and defined as follows.
Consider the F-vector space V1 ⊗ V2 (this is the ordinary tensor product
of vector spaces; again we have the same notation for different objects, but
the context always clarifies whether we are dealing with the vector space or
2
with the module). Given the bases {v11 , . . . , vn1 } and {v12 , . . . , vm
}, for V1
12
1. SOME FUNDAMENTAL CONCEPTS
and V2 respectively, an action of G on each element of the kind vi1 ⊗ vj2 can
be defined in a componentwise fashion, by
(vi1 ⊗ vj2 )g := (vi1 )g ⊗ (vj2 )g
for all i, j in the relevant sets of indices, and for all g in G. Then we
extend these relations by F-linearity to the whole V1 ⊗ V2 ; it is not hard to
check that the structure of FG-module on V1 ⊗ V2 which indeed arises in
this way depends only (up to isomorphism) on the modules V1 and V2 , and
not on the choice of bases.
Inner tensor product and induction of modules are nicely related to each
other, in a way which is expressed by the first part of the following theorem
(see [HB], VII, 4.15); as a consequence of it, the second part provides useful
informations on what happens when a module is restricted and then induced
back.
1.23. Theorem. Let G be a group, H a subgroup of G, F a field, V an
FG-module, and W an FH -module. Then (V ↓H ⊗W ) ↑G ' V ⊗ (W ↑G )
holds. In particular, denoting by X the 1-dimensional trivial FH -module,
we have (V ↓H )↑G ' V ⊗ (X↑G ).
The second part of the previous theorem turns out to be particularly
useful when H is a normal subgroup of G; indeed, in this case we have
X↑G ' F[G/H], where F[G/H] is the (right) regular module of G/H over
F viewed in an obvious way as an FG-module.
1.24. Definition. Let H and K be groups, F a field, and W , Z modules
over F for H and K respectively; the outer tensor product of W and Z is
an F[H ×K]-module, whose dimension is (dim W )(dim Z), which is denoted
by W # Z and defined as follows. Consider the F-vector space W ⊗ Z , and
the basis {wi ⊗ zj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} for it (which arises as usual from
two given bases, {w1 , . . . , wn } and {z1 , . . . , zm }, for W and Z respectively);
we define an action of H × K on W ⊗ Z (which yields the module W # Z )
extending by F-linearity the relations
(wi ⊗ zj )(h,k) := wih ⊗ zjk .
As in the case of inner tensor products, the construction above is not affected, up to isomorphism, by the choice of bases.
Observe that, given a group G, there exists an obvious monomorphism
δ from G to G × G defined by gδ := (g, g) for all g in G; in this way G is
TENSOR PRODUCTS AND TENSOR INDUCTION OF MODULES
13
realized as a ‘diagonal’ of G × G. Now, if V1 and V2 are FG-modules, it is
clear that the inner tensor product V1 ⊗V2 can be recognized as (V1 #V2 )↓Gδ .
We move now to introduce the concept of tensor induction for modules,
which will be really central in our discussion. As it was mentioned in the
introduction, tensor induction can be viewed in many senses as a faithful
transposition, to a multiplicative context, of (ordinary) induction: if G is
a group, H a subgroup of G, F a field, and W an FH -module, tensor
induction provides a method for building an FG-module, whose dimension
is (dim W )|G:H| . One of the main analogies between the two concepts of
induction lies in the fact that they can be defined in a virtually identical
way (roughly speaking, we have only to change the symbols ‘⊕’ and ‘+’ with
the symbol ‘⊗’). But we should rather say that this holds if we make the
choice of defining ordinary induction in the way we did it; for example, there
seems to be no chance of giving a choice-free definition for tensor induction
(see 1.10). At any rate, the most important difference between the two
processes lies probably in the fact that, while ordinary induction may be
characterized as adjoint to restriction (on both sides), so far nothing similar
is known for tensor induction.
1.25. Definition. Let G be a group, H a subgroup of G whose index is n,
F a field, and W an FH -module. We construct an FG-module, denoted by
W↑⊗G and such that dim W↑⊗G = (dim W )n , as follows: we repeat step (a)
of 1.9, but taking W ⊗n (which is the tensor product of n copies of W )
as the underlying vector space, instead of W ⊕n ; in this way of course the
underlying vector space has the desired dimension. The actions of H n and
Sn on W ⊗n are the linear extensions of
h
(w1 ⊗ · · · ⊗ wn )(h1 ,...,hn ) := w1 1 ⊗ · · · ⊗ wnhn
and
(w1 ⊗ · · · ⊗ wn )ξ := w1ξ−1 ⊗ · · · ⊗ wnξ−1 ,
and together they define an action of H o Sn on W ⊗n . As in 1.9(b), the
composite of a Frobenius embedding of G in H oSn with this action may now
be taken to define an action of G. The FG-module so obtained is denoted
by W↑⊗G . Finally, we see that the construction above is well defined up to
isomorphism (the same observations that we made in Remark 1.10 are valid
in the present context), and we say that any FG-module which is isomorphic
to W↑⊗G is tensor-induced by W from H .
14
1. SOME FUNDAMENTAL CONCEPTS
We finally remark that a statement like 1.11, concerning transitivity,
holds perfectly for tensor induction too.
Representations and characters
The concept of representation is very deeply related to the concept of
module; indeed, we could say that they provide two different languages for
describing essentially the same picture. For a large part of our discussion the
‘module approach’ will turn out to be the convenient one, but in Chapters
2 and 3, while we shall be facing directly a multiplicative context, we can
not avoid to deal with representations rather than with modules. This is
mainly because, as we shall see, ordinary Representation Theory is not the
appropriate environment for studying the multiplicative structure of representations; in this kind of analysis, it is necessary to introduce the concept
of projective representation (see next section), which appears to be of much
larger use, in the literature, than the corresponding concept of ‘premodule’.
For this reason, we briefly recall here what representations are, how they are
linked to modules, and how several concepts introduced in earlier sections
can be translated in the language of representations.
1.26. Definition. Let G be a group, F a field, and n a positive integer.
(a) If D is a group homomorphism from G to the general linear group
GL(n, F), then we say that D is an F-representation of degree n for
G (most of the time there is no need to give such an emphasis, in the
notation, to the field F). If the kernel of D is trivial, then we say that
D is a faithful representation; in this case, of course, D provides an
embedding of G in GL(n, F).
(b) Let D1 and D2 be F-representations of the same degree for G. If there
exists a matrix A in GL(n, F) such that D2 (g) = A−1 D1 (g)A holds for
all g in G, then we say that D1 and D2 are equivalent. This defines an
equivalence relation on the set of F-representations for G.
Consider now an n-dimensional vector space V over F. Given a basis
for V , a group isomorphism between AutF (V ) and GL(n, F) arises immediately; therefore, a given F-representation of degree n for G yields a
structure of FG-module on V , and it is easily seen that the choice of the
basis does not affect this process up to isomorphism. Conversely, if a structure of FG-module is defined on V , then an F-representation of degree n
for G arises as soon as a basis for V is chosen, and different choices of
REPRESENTATIONS AND CHARACTERS
15
the basis lead to equivalent representations. Moreover, isomorphic module
structures yield equivalent representations and vice versa, so that we have
a well defined (one to one) correspondence between isomorphism types of
modules and equivalence classes of representations.
Without giving the ‘real’ definitions, we can say that a representation
is irreducible if and only if it is associated to a simple module (similarly for
absolute irreducibility), whereas it is completely reducible if and only if it
is associated to a semisimple module. If D is a completely reducible representation and V is a semisimple module associated to it, the irreducible
constituents of D (which are defined up to equivalence) are the representations associated to the simple constituents of V . Also, in this setting, D
is called homogeneous if its irreducible constituents are pairwise equivalent,
and this of course happens if and only if V is homogeneous.
We shall go into a little more details introducing the concept of induced representation. For doing that, we need some ingredients: let G be
a group, H a subgroup of G of index n, F a field, and T a representation for H . We denote by T o Sn the ‘obvious’ homomorphism from H o Sn
to GL(deg T, F) o Sn which maps an element (h1 , . . . , hn )π to the element
(T (h1 ), . . . , T (hn ))π . Also, we denote by m the so called ‘block monomial
embedding’ of GL(deg T, F) o Sn in GL(n deg T, F) (m is the monomorphism of groups which maps an element (A1 , . . . , An ) of the base group to
(A1 ⊕ · · · ⊕ An ), where A1 ⊕ · · · ⊕ An is the block-diagonal matrix whose diagonal blocks are the matrices Ai ; an element π of the top group is mapped
by m to a permutation matrix in GL(n deg T, F) which acts by conjugation,
on the image of the base group, permuting the blocks as π permutes the
indices). Finally, let ϕ be a Frobenius embedding of G in H o Sn .
1.27. Definition. Let G be a group, H a subgroup of G of index n, F a
field, and T an F-representation for H . Then we define the map
T↑G : G → GL(n deg T, F)
as the composite map ϕ(T o Sn )m. We see that T↑G is an F-representation
for G (of degree n deg T ), and the equivalence type of T ↑G is not affected by any of the choices involved in the defining process; also, if R is
an F-representation equivalent to T , then R ↑G is equivalent to T ↑G as
well. We say that any F-representation of G which is equivalent to T↑G is
induced by T from H .
16
1. SOME FUNDAMENTAL CONCEPTS
As we expect, if W is an FH -module associated to T , then W ↑G is
associated to T↑G . Also, an irreducible representation of a group G is called
primitive if it is not induced by any proper subgroup of G, and clearly this
happens if and only if it is associated to a primitive module.
The concept of restriction for representations is very natural and does
not require any definition (a representation is a map, and it is very well
known what restriction of maps is); now, an irreducible representation of a
group G is called quasi-primitive if its restriction to any normal subgroup
of G is homogeneous. There is no need to explain that, if H is a subgroup
of G, D is a representation of G, and V is a module associated to D , then
D↓H is associated to V ↓H , and D is quasi-primitive if and only if V is so.
Also the concepts of inner tensor product, outer tensor product and,
above all, tensor induction, deserve a translation in the language of representations; before giving the relevant definitions, it is worth recalling what
the Kronecker product of matrices is. Let A = (aij ) and B = (bkl ) be
matrices, say of size m × n and s × t respectively, with entries in a field (or
also in a ring); the Kronecker product of A and B is a matrix, denoted by
A ⊗ B , which has size ms × nt and which can be viewed as a block matrix,
having the block aij (bkl ) = aij B in position (i, j).
1.28. Definition. Let G be a group, F a field, and D1 , D2 F-representations for G having degrees n and m. The map D1 ⊗ D2 : G → GL(nm, F)
defined by
(D1 ⊗ D2 )(g) := D1 (g) ⊗ D2 (g)
for all g in G, is an F-representation for G which is called the inner tensor
product of D1 and D2 .
1.29. Definition. Let H and K be groups and F a field. Consider
an F-representation R of degree n for H and an F-representation S of
degree m for K . The map R # S : H × K → GL(nm, F) defined by
(R # S)(h, k) := R(h) ⊗ S(k)
for all (h, k) in H × K , is an F-representation for H × K which is called
the outer tensor product of R and S .
As we have seen in the context of modules, tensor induction for representations can be defined in deep analogy with ordinary induction. Recalling the setting of 1.27, in the following definition we will make use of a
Frobenius embedding ϕ, and of the map T o Sn , exactly as we did in that
REPRESENTATIONS AND CHARACTERS
17
situation; there is only need to replace the block monomial embedding with
the homomorphism k : GL(deg T, F) o Sn → GL((deg T )n , F) which maps
an element (A1 , . . . , An ) of the base group to the element (A1 ⊗ · · · ⊗ An ),
whereas an element π of the top group is mapped to a permutation matrix
in GL((deg T )n , F) which acts by conjugation, on the image of the base
group, permuting the Kronecker factors as π permutes the indices.
1.30. Definition. Let G be a group, H a subgroup of G of index n, F a
field, and T an F-representation for H . Then we define the map
T↑⊗G : G → GL((deg T )n , F)
as the composite map ϕ(T o Sn )k . We see that T↑⊗G is an F-representation
for G of degree (deg T )n , and the equivalence type of T ↑⊗G is not affected by any of the choices involved in the defining process; also, if R is
an F-representation equivalent to T , then R↑⊗G is equivalent to T ↑⊗G as
well. We say that any F-representation of G which is equivalent to T ↑⊗G
is tensor-induced by T from H .
We conclude this subsection on representations with the following, not
at all surprising,
1.31. Remark. Let G be a group, F a field, and D1 , D2 F-representations
of G associated to the modules V1 and V2 respectively. Then the representation D1 ⊗ D2 is associated to the inner tensor product V1 ⊗ V2 (observe
that this provides an easy way to conclude that the inner tensor product of
two modules depends only, up to isomorphism, on the tensor factors). A
similar feature holds for outer tensor products; also, if H is a subgroup of
G, and T is an F-representation for H associated to the module W , then
T↑⊗G is associated to W↑⊗G .
From time to time we shall find convenient to use the language of characters; here we recall briefly some basic definitions and properties, concerning
characters over the complex field, which will be useful in the sequel.
1.32. Definition. Let G be a group, and D a complex representation of G.
(a) The map χ : G → C, defined by
χ(g) := tr(D(g))
for all g in G, is called the character of G afforded by D (here tr(D(g))
denotes the trace of the matrix D(g)). Observe that the value χ(1) gives
the degree of D , which is also, by definition, the degree of χ.
18
1. SOME FUNDAMENTAL CONCEPTS
(b) If D is an irreducible representation, then the character χ afforded by D
is called irreducible as well. We denote by IrrG the set of all irreducible
complex characters of G.
(c) The kernel of the character χ afforded by D is defined as the set
ker χ := {g ∈ G : χ(g) = χ(1)}. We say that χ is faithful if its
kernel is trivial (it is not hard to check that ker χ is the same as ker D ,
hence χ is faithful if and only if D is so).
1.33. Remark. Let G be a group.
(a) If χ is a complex character of G, then it is easily seen that χ is a class
function; this means that χ takes the same value on elements which are
conjugate in G. Indeed, it can be shown that the set of all the class
functions from G to C is, in an obvious way, a vector space over C
whose dimension is the number of conjugacy classes of G, and the set
IrrG provides a basis for it.
(b) Let χ1 and χ2 be complex characters of G, afforded by the representations D1 and D2 respectively; then it can be shown that D1 and D2
are equivalent if and only if χ1 = χ2 .
(c) Let ψ be the complex character of G afforded by the representation D .
Then there exists a unique sequence of nonnegative integers (not all
zero) {aχ : χ ∈ IrrG}, such that we have
X
ψ=
aχ χ.
χ∈IrrG
Each χ in IrrG such that aχ 6= 0 is called an irreducible constituent
of ψ . Of course, the irreducible constituents of ψ are precisely the
characters afforded by the irreducible constituents of D .
(see [Is], Chapter 2).
1.34. Definition. Let G be a group, H a subgroup of G, and ϑ a complex
character of H . For any g in G, we define the map ϑg : H g → C by setting
−1
ϑg (y) := ϑ(y g )
for all y in H g . It is easily seen that ϑg is a character of H g and, if W
is the CH -module which affords ϑ, then ϑg is afforded by the translate
W g of Definition 1.16. If H is a normal subgroup, then we can define the
set IG (ϑ) := {g ∈ G : ϑg = ϑ}; it is clear that IG (ϑ) coincides with the
subgroup IG (W ) defined in the preamble before 1.19, and it is called the
inertia subgroup of ϑ in G.
PROJECTIVE REPRESENTATIONS
19
1.35. Definition. Let G be a group, N a normal subgroup of G, and χ a
character of G/N . Then the map χinf : G → C defined by
χinf (g) := χ(N g)
for all g in G, is a character of G, which is said to be obtained by inflation
from χ. It is easy to see that χinf is irreducible if and only if χ is so.
All the terminology introduced for representations and modules can be
translated in an obvious way in the language of characters; for example, we
shall speak of quasi-primitive characters, induced characters, and so on.
Projective representations
In the course of the previous sections we have seen that, given an irreducible representation for a group G, a combination of Clifford’s Theorem
and induction provides a very effective algorithm for detecting a ‘convenient’ quasi-primitive representation, where convenient means that we know
exactly how to reconstruct our original representation from it. For this reason, in studying completely reducible representations, we can actually focus
on quasi-primitive representations. It was also mentioned that the additive
structure of a quasi-primitive representation is no more helpful (at least, we
don’t have any general method to exploit it) in order to reduce our analysis further; we are indeed approaching a context in which ‘multiplicative
methods’ are needed (this will be actually the starting point for this thesis).
Roughly speaking, one may ask whether a given quasi-primitive representation D can be realized as the inner tensor product of representations
of smaller degree, and at this stage it is appropriate to introduce projective
representations; these objects, which arise by generalizing the usual concept
of representation, turn out to be relevant here because it makes perfectly
sense to construct inner tensor products with them, and such a product
may happen to be a representation in the classical sense (sometimes, for
the sake of emphasis, a representation in the classical sense will be called a
genuine representation). Hence, if we don’t want to overlook a significant
aspect of the picture, we have to take in account that D could admit tensor
factorizations in which the factors are projective representations.
Before going through the discussion, we remark that the concept of projective representation reveals some very interesting connections with cohomology theory. Also, it should be clarified that projective modules in the
20
1. SOME FUNDAMENTAL CONCEPTS
sense of Homological Algebra (which are important objects in Representation Theory) have nothing to do with this context; as we mentioned, sometimes in the literature the concept of ‘premodule’ is paired with projective
representations (for example, in [Be2], 1, B), but this does not seem to be
of very common use. An exhaustive discussion about projective representations (including proofs for many results stated in this section) can be found
in [Is], Chapter 11.
1.36. Definition. Let G be a group, F a field, and n a positive integer.
(a) Let P be a map from G to GL(n, F). If there exists a map α, from
G × G to F, such that P (g1 )P (g2 ) = α(g1 , g2 )P (g1 g2 ) holds for all g1 ,
g2 in G, then P is called a projective F-representation of degree n for
G (sometimes the reference to the field F is dropped).
(b) Let P1 and P2 be projective F-representations of degree n for G. If
there exist a matrix A in GL(n, F) and a map β : G → F such that
we have P2 (g) = β(g)A−1 P1 (g)A for all g in G, then we say that P1
and P2 are equivalent. This defines an equivalence relation on the set
of projective representations of G, and we shall denote by [P ] the class
of a projective representation modulo this equivalence relation.
(c) Let P be a projective F-representation of degree n for G, and V an
n-dimensional vector space over F. We say that P is irreducible if the
only subspaces of V which are invariant under the action of P (G) are
the zero space and V itself.
From the definition above, it is clear that a projective representation P
fails (in general) to be a genuine one just because it is ‘disturbed’ by the
map α, which is called the factor set associated to P . Since P takes values
in a general linear group, we immediately see that α is indeed a map from
G×G to the multiplicative group F× of F, and it is uniquely determined by
P . Observe that, for similar reasons, the map β of 1.36(b) does not vanish
on any element of G.
It is easily seen that, for all x, y, z in G, we get
α(xy, z)α(x, y) = α(x, yz)α(y, z),
and this is actually a characterizing property for factor sets, in the sense
that any map from G × G to F× which satisfies this property turns out
to be the factor set of some projective representation for G. The set of all
F× -factor sets of G constitutes a group (with respect to pointwise multiplication) which is denoted, in the language of group cohomology, by Z 2 (G, F× )
PROJECTIVE REPRESENTATIONS
21
(the group of 2-cocycles). Now, the second cohomology group H 2 (G, F× )
can be realized as a quotient of Z 2 (G, F× ) (modulo the relevant subgroup
of 2-coboundaries B 2 (G, F× )), and it can be seen that equivalent projective representations yield factor sets which represent the same element of
H 2 (G, F× ).
1.37. Remark. If P̄ is the composite of a projective F-representation P
of degree n for G with the natural homomorphism π which maps GL(n, F)
onto P GL(n, F), then P̄ is a homomorphism from G to P GL(n, F).
Conversely, if τ : G → P GL(n, F) is a homomorphism, a projective
F-representation of degree n for G arises in a natural way (as soon as
we choose a transversal for Z(GL(n, F)) in the full preimage under π of
τ (G); such a choice is however not relevant up to equivalence). For this reason we can say that a projective F-representation of degree n for G reflects
an action of G on a projective space of dimension n − 1 over F.
Now, we introduce a concept of equivalence for homomorphisms to projective general linear groups: let τ1 and τ2 be homomorphisms of G to
P GL(n, F); we say that τ1 and τ2 are equivalent if there exists A in
GL(n, F) such that τ2 (g) = τ1 (g)(π(A)) for all g in G (and, when we write
the symbol ‘'’ between two such homomorphisms, we refer to this kind of
equivalence). If we choose now two projective representations P1 and P2 of
G with P¯1 = τ1 and P¯2 = τ2 , it is clear that τ1 and τ2 are equivalent if
and only if P1 and P2 are so.
As we mentioned in the preamble of this section, projective representations are important for us because it is possible to construct inner tensor
products (and, similarly, outer tensor products) with them.
1.38. Definition. Let G be a group, F a field, and P1 , P2 projective
F-representations for G, with factor sets α1 , α2 and degrees n, m respectively; then the map P1 ⊗ P2 : G → GL(nm, F) defined by
(P1 ⊗ P2 )(g) := P1 (g) ⊗ P2 (g)
for all g in G, is a projective F-representation of G whose factor set is
the product (in Z 2 (G, F× )) of α1 and α2 ; this projective representation is
called the inner tensor product of P1 and P2 .
It is now clear that the inner tensor product of two projective representations may yield a genuine representation; this happens exactly when the
relevant factor sets are inverse of each other in Z 2 (G, F× ).
22
1. SOME FUNDAMENTAL CONCEPTS
Also the concept of tensor induction can be extended to projective representations; in the following definition we shall make use of the same notation
as in 1.30:
1.39. Definition. Let G be a group, H a subgroup of G of index n,
F a field, and P a projective F-representation for H . Then the map
P ↑⊗G : G → GL((deg P )n , F) defined as the composite map ϕ(P o Sn )k
is a projective F-representation of G (of degree (deg P )n ). Any projective
F-representation of G which is equivalent to P ↑⊗G is said to be tensorinduced by P from H .
We can now introduce some more notation.
1.40. Definition. Let G be a group, H a subgroup of G of index n, and
F a field.
(a) If P1 and P2 are projective F-representations of G, then we denote by
P¯1 ⊗ P¯2 the homomorphism P1 ⊗ P2 .
(b) If P is a projective F-representation of H , then we denote by P̄ ↑⊗G
the homomorphism P↑⊗G .
1.41. Remark. The good behaviour of projective representations, with respect to multiplicative constructions, lies essentially in the fact that the
homomorphism from GL(n, F) × GL(m, F) to GL(nm, F) given by the
Kronecker product does yield a homomorphism (which is injective) from
P GL(n, F) × P GL(m, F) to P GL(nm, F). Indeed, following this line, it is
possible to define the concepts of inner tensor product and tensor induction
for homomorphisms to projective general linear groups (in [Kov], by definition, a projective representation is a homomorphism of this kind); without
going into details, we just mention that Definition 1.40 turns out to be
perfectly consistent with this approach.
A similar situation does not hold for the block monomial embedding
of GL(n, F) × GL(m, F) to GL(n + m, F), and for this reason there is not
a sensible way to define ‘sums’, or ordinary induction, in the context of
projective representations.
Since any genuine representation is also a projective representation, some
confusion could arise from the fact that two different concepts of equivalence
are floating around; we will emphasize this distinction, when needed, saying
that two representations are genuine-equivalent if they are equivalent in the
classical sense of 1.26(b), whereas we shall call them projective-equivalent
PROJECTIVE REPRESENTATIONS
23
if they are equivalent in the sense of 1.36(b). At any rate, the following
remark is a useful one.
1.42. Remark. If D1 and D2 are genuine representations of the group G
which are projective-equivalent, they are not necessarily genuine-equivalent;
indeed, it is straightforward to check that D1 is projective-equivalent to
D2 if and only if there exists a 1-dimensional representation λ of G such
that D1 is genuine-equivalent to λ ⊗ D2 . Observe that every map from
G to F× is a projective representation of degree 1, and all these projective
representations of G are equivalent. As another example, the two irreducible
2-dimensional C-representations of S3 × C2 are projective-equivalent.
We conclude this section presenting the statement of an important theorem of Schur, which shows that any projective complex representation of a
given group G can be ‘realized’ as a genuine representation of a larger group
(which is finite, and whose structure is linked to the cohomology theory of
G). For this purpose, we first introduce the concept of central extension.
1.43. Definition. Let G be a group; a central extension of G is a pair
(Γ, π), where Γ is a (possibly infinite) group, and π is a homomorphism of
Γ onto G such that ker π lies in Z(Γ). If the group Γ is finite, then (Γ, π)
is called a finite central extension.
1.44. Definition. Let (Γ, π) be a finite central extension of a group G. We
say that (Γ, π) has the projective lifting property for G if, for any projective
C-representation P of G, there exists a genuine C-representation D of Γ,
together with a map µ : Γ → C× , such that we have
D(g) = P (π(g))µ(g)
for all g in G.
Now we are in a position to state the promised theorem of Schur.
1.45. Theorem. For any group G, there exists a finite central extension
(Γ, π) which has the projective lifting property for G. Moreover, (Γ, π) can
be chosen such that ker π is isomorphic to H 2 (G, C× ) and ker π lies in Γ0 .
The second cohomology group H 2 (G, C× ) is also known as the Schur
multiplier of G, and any finite central extension given by Theorem 1.45 is
called a Schur covering for G.
CHAPTER 2
On tensor factorization
As the discussion in the previous chapter suggests, the analysis of quasiprimitive representations from a ‘multiplicative’ point of view turns out
to be a relevant issue in Representation Theory; here we start moving in
that direction, focusing on the classical context of representations over the
complex field (although very soon we shall be dealing with modules over
finite fields); at any rate, what is needed in the following discussion is to
work over an algebraically closed field of characteristic zero.
Our first aim, which will be pursued in this chapter, is to achieve a good
understanding on how a given quasi-primitive representation of a group G
can ‘split’ into the inner tensor product of two projective representations.
More precisely, assume that D is a (faithful) quasi-primitive representation
of G, with the further hypothesis that the restriction of D to the Fitting subgroup F of G is irreducible; the main result of this chapter (Corollary 2.10)
shows essentially that there is a bijection between the set of all the pairs
([P1 ], [P2 ]), where P1 and P2 are projective representations of G such that
D is projective-equivalent to P1 ⊗ P2 , and the set of normal subgroups of
G lying between the centre Z and F . Thus, we get a parametrization of
all the possible ways in which (the projective-equivalence class of) D can
be factorized, in terms of the group structure of G. The assumption of
faithfulness for D was added here just for comfort, whereas the hypothesis
that D↓F is irreducible appears to be a natural one, especially in a solvable
context. Indeed, if D↓F is not irreducible, then there are two possibilities:
either D can be already factorized by means of a well known result (which
we state below as Lemma 2.1), or the Fitting subgroup of G coincides with
the centre; if G is solvable, the second option implies that G is abelian
(indeed cyclic, if D is faithful), and the complex representation theory of a
finite abelian group is not an issue.
Since the result sketched above describes essentially how the projectiveequivalence class of D factorizes, it can be conveniently translated in the
language of characters, and this is done by means of Theorem 2.12. Finally,
25
26
2. ON TENSOR FACTORIZATION
Remark 2.13 shows that the hypothesis of irreducibility for D↓F can not be
weakened (if we don’t want to spoil the results) even if D is assumed to be
primitive instead of quasi-primitive.
In what follows we shall make large use of notation and concepts introduced in the last section of Chapter 1; also, we shall denote by Ie (e being
a positive integer) the trivial complex representation of degree e, as well
as the e-dimensional identity matrix over C. As a last remark, the generic
term of representation will always stand for ‘C-representation’.
Factorization of representations
The main result, which is 2.10, follows as a corollary of Theorem 2.8.
Before proving it, we need to prepare the setting with some preliminary
lemmas (2.7 turns out to be particularly useful); the first one, which is only
stated (see [Hu2], 21.1 and 21.2 for a proof), is also known as the ‘Second
Theorem of Clifford’.
2.1. Lemma. Let G be a group, N a normal subgroup of G, T an irreducible representation of N , and D an irreducible representation of G such
that D↓N = Ie ⊗ T . Then there exist projective representations P1 and P2
of G such that
(a) D(g) = P1 (g) ⊗ P2 (g) for all g in G,
(b) P1 (x) = Ie and P2 (x) = T (x) for all x in N ,
(c) T (x)P2 (g) = T (xg ) for all x in N and g in G.
2.2. Remark. Let G be a group, N a normal subgroup of G, and T an
irreducible representation of N ; assume also that P and P 0 are projective
0
representations of G such that T (x)P (g) = T (xg ) = T (x)P (g) holds for all
x in N and g in G, that is, P and P 0 both fulfill condition (c) of 2.1.
Then, by Schur’s Lemma (1.3), we get P̄ = P¯0 (here we use the notation
introduced in 1.37) and we conclude that, in the context of Lemma 2.1, N
and D determine uniquely P¯2 .
Suppose now that T 00 is a representation of N which is equivalent to T ,
and that P 00 satisfies condition (c) of 2.1 with respect to T 00 , whereas P still
satisfies that condition with respect to T ; then it is clear that P and P 00 are
projective-equivalent. Therefore we can say that, in the context of 2.1 (but
changing ‘=’ in the assumption and in conclusion (a) with ‘'’), N and the
genuine-equivalence type of D determine uniquely the projective-equivalence
type of P2 .
FACTORIZATION OF REPRESENTATIONS
27
2.3. Remark. Let the group G be the direct product of the subgroups A
and B ; if R is a representation of A and S a representation of B , then we
know that the outer tensor product R#S is a representation of G (see 1.29).
It is not hard to check (see [Is], 4.21) that, if R and S are irreducible, then
R # S is also irreducible; moreover, each irreducible representation D of G
is equivalent to the outer tensor product of an irreducible representation of
A and an irreducible representation of B , and the equivalence type of D
determines uniquely the equivalence types of both these representations.
Suppose now that G is a central product of A and B , which means that
we have G = AB with [A, B] = 1, so that A ∩ B ≤ Z(G). In this case G
is a quotient of the (external) direct product A × B (see [Gor], 3.7.2), and
by inflation each representation of G may be viewed as a representation of
A × B . In particular, each irreducible representation of G may be viewed as
an outer tensor product R#S of some irreducible representation R of A and
some irreducible representation S of B , such that R↓A∩B = ξ ⊗ Ideg R and
S↓A∩B = ξ ⊗ Ideg S for a suitable 1-dimensional representation ξ of A ∩ B .
Conversely, it is easy to see that, if R and S are irreducible representations
of A and B satisfying this condition for some ξ , then R # S may be viewed
as an (irreducible) representation of G. We shall take full advantage of these
facts, and write simply ‘is’ instead of ‘may be viewed as’.
2.4. Lemma. Let the group G be a central product of the subgroups A and
B . Then the following properties hold:
(a) if R is an irreducible genuine representation for one of the central factors, say A, then there exists a unique homomorphism %, from G to
P GL(deg R, C), such that %↓A = R̄ and B ≤ ker %,
(b) let R and S be irreducible genuine representations, for A and B respectively, such that R # S is a representation for G. If % and σ are
homomorphisms as in (a) for R and S , then we have R # S = % ⊗ σ .
Proof of (a). For each element g of G we can find an element a in
A and an element b in B such that g = ab; now, we define %(g) to be
R̄(a). Observe that the value %(g) does not depend on the choice of the
decomposition of g : if a1 is in A, b1 is in B and g = ab = a1 b1 , then
a−1 a1 = bb−1
is an element of A ∩ B , and therefore it lies in the centre
1
of G; hence we get (R̄(a))−1 R̄(a1 ) = R̄(a−1 a1 ) = 1. This tells us that
% is a well defined map from G to P GL(deg R, C), which is obviously a
homomorphism since R̄ is. It is straightforward that %↓A is R̄ and that B
28
2. ON TENSOR FACTORIZATION
is contained in ker %; also, these two conditions yield the uniqueness part of
the statement.
Proof of (b). Let X and Y be projective representations of G such
that X̄ = % and Ȳ = σ (see Remark 1.37); following Definition 1.40, % ⊗ σ
is the homomorphism X ⊗ Y . Now, given an element g in G, let a and
b be elements of A and B respectively, such that g = ab; since we get
X̄(g) = %(g) = R̄(a), and Ȳ (g) = σ(g) = S̄(b), claim (b) follows simply by
applying the definitions.
Now we introduce a class of groups which will play an important role;
then, in Lemma 2.7, several properties of these groups are outlined.
2.5. Definition. Let F be a finite group; we say that F is a good group if
Z(F ) is cyclic and F/Z(F ) is abelian of squarefree exponent.
The next result, concerning good groups, will be useful for us; see [FT],
1.4 for a proof of it.
2.6. Lemma. Let G be a group with Fitting subgroup F and centre Z .
If G has a faithful quasi-primitive representation, then F/Z is an abelian
group of squarefree exponent.
2.7. Lemma. Let F be a good group, and let Z denote its centre; then the
following properties hold:
(a) if K is a subgroup of F such that Z(K) = Z , then F is the (central)
product of K and CF (K);
(b) if P is an irreducible projective representation of F with Z ≤ ker P̄ ,
then we have (deg P )2 = |F : ker P̄ |;
(c) if D is a faithful irreducible representation of F , and D̄ ' P¯1 ⊗ P¯2
where P1 and P2 are projective representations of F (here equivalence
is in the sense of Remark 1.37), then we have F = ker P¯1 · ker P¯2 ;
(d) with the same assumptions as in (c), if K is the kernel of P¯1 , then
Z(K) coincides with Z ; moreover, denoting by L the kernel of P̄2 , we
have L = CF (K);
(e) with the same assumptions as in (c), there exist genuine representations
R and S , of K := ker P¯1 and L := ker P¯2 = CF (K) respectively,
such that R # S is a representation of F which is genuine-equivalent
to D ; moreover, we have P¯1 = σ and P¯2 = %, where % and σ are the
homomorphisms linked to R and S , which were constructed in 2.4(a),
so that we have P¯1 ↓L = S̄ and P¯2 ↓K = R̄. This implies that P1 ↓L is a
FACTORIZATION OF REPRESENTATIONS
29
genuine representation of L up to multiplying it by a suitable map from
L to C× , and P2 behaves similarly with respect to K .
Proof of (a). Let Q be a (nontrivial) Sylow subgroup of F , say for the
prime q . We claim first that Z(Q) = Z(K ∩ Q). To see this, note that
Z(Q) ≤ Z because F is nilpotent; as Z ≤ K , this proves that Z(Q) is
contained in K ∩ Q, and then of course Z(Q) ≤ Z(K ∩ Q). Conversely,
Z(K ∩ Q) lies in Z(K) = Z because K is nilpotent and K ∩ Q is the Sylow
q -subgroup of K , and hence Z(K ∩ Q) ≤ Z(Q).
It follows that (K ∩ Q) ∩ CQ (K ∩ Q) = Z(Q), whence the product of
(K ∩ Q)/Z(Q) and CQ (K ∩ Q)/Z(Q) is a direct product. We want to show
next that this is is all of Q/Z(Q). Suppose that |(K ∩ Q)/Z(Q)| = q n so
(K ∩ Q)/Z(Q) is an n-dimensional vector space over GF (q), and choose a
T
basis {Z(Q)x1 , . . . , Z(Q)xn } for it. We have CQ (K ∩ Q) = ni=1 CQ (xi ).
Using that F/Z is abelian of squarefree exponent, it is easy to see that each
map αi : Q → Z defined by αi (x) = [xi , x] is a homomorphism whose kernel
is CQ (xi ) and whose image has exponent dividing q . Since Z is cyclic, we
conclude that |Q : CQ (xi )| ≤ q , whence |Q/CQ (K ∩ Q)| ≤ q n . Thus the
dimension of (K ∩ Q)/Z(Q) × CQ (K ∩ Q)/Z(Q) (as a vector space over
GF (q)) is at least the dimension of Q/Z(Q), and our claim follows.
Finally, let {Q1 , . . . , Qh } be the set of (nontrivial) Sylow subgroups
of F : we have
F = (K ∩ Q1 )CQ1 (K ∩ Q1 ) · · · (K ∩ Qh )CQh (K ∩ Qh ) = KCF (K)
as we wanted.
Proof of (b). Since P is an irreducible projective representation such
that ker P̄ contains Z , we have that P̄ (F ) is an irreducible abelian subgroup of P GL(deg P, C) (which is of course isomorphic to F/ ker P̄ ); if
we denote by M the preimage of P̄ (F ) in GL(deg P, C) under the natural homomorphism, we have Z(GL(deg P, C)) ≤ Z(M ) but, since M is
irreducible, equality holds. Moreover, M is nilpotent of class 2, so that
(deg P )2 = |M/Z(M )| = |P̄ (F )| (this is not hard to prove; see for example
[Dix], 4.3) and our claim follows.
Proof of (c). Since D is faithful, we have ker P̄1 ∩ ker P̄2 = Z (this is
easily seen, taking in account that the Kronecker product of two matrices
is a scalar matrix if and only if the factors are scalar matrices), so that
ker P̄1 /Z · ker P̄2 /Z = ker P̄1 /Z × ker P̄2 /Z ; this is a subgroup of the abelian
30
2. ON TENSOR FACTORIZATION
group with squarefree exponent F/Z , hence it suffices to show that
l(ker P̄1 /Z) + l(ker P̄2 /Z) = l(F/Z)
(where by l(G) we mean the composition length of the group G).
Now, l(ker P̄i /Z) = l(F/Z) − l(F/ ker P̄i ), so we want to prove that
l(F/Z) = l(F/ ker P̄1 ) + l(F/ ker P̄2 )
holds. We have F/Z ' D̄(F ) and F/ ker P̄i ' P̄i (F ); let H be the set
{(P¯1 (x), P¯2 (x)) : x ∈ F }, which is indeed a subgroup of the (external)
direct product P¯1 (F ) × P¯2 (F ), and let ϕ : H → P GL(deg D, C) be the map
defined by
ϕ((P¯1 (x), P¯2 (x))) = P1 (x) ⊗ P2 (x).
We know that ϕ is a monomorphism (recall 1.41), and therefore
H ' ϕ(H) ' D̄(F )
holds; but now, by part (b), we have
|D̄(F )| = (deg D)2 = (deg P1 )2 (deg P2 )2 = |P̄1 (F )||P̄2 (F )|,
whence D̄(F ) ' P̄1 (F ) × P̄2 (F ), and the claim is proved.
Proof of (d). By part (c) we have F = KL. Let us now prove that
[K, L] = 1. Denoting by x an element of K and by y an element of L,
there exist A in GL(deg D, C) and λ, µ in C× such that
A−1 D(x)A = λIdeg P1 ⊗ P2 (x) and A−1 D(y)A = P1 (y) ⊗ µIdeg P2 ;
it is now clear that [D(K), D(L)] = 1, and the faithfulness of D yields what
we wanted. Of course now we have Z(K) = Z and, since the conditions
F = KL, L ≤ CF (K) and K ∩ L = K ∩ CF (K) = Z hold, we conclude
that L = CF (K).
Proof of (e). By assumption, we can find an element A in GL(deg D, C)
and a map λ from F to C× such that λ(f )A−1 D(f )A = P1 (f ) ⊗ P2 (f )
holds for all f in F . Now, for all k in K , the previous equality becomes
λ(k)A−1 D(k)A = µ(k)Ideg P1 ⊗ P2 (k), where µ is a map from K to C× ;
defining R(k) as λ(k)−1 µ(k)P2 (k) we get A−1 D(k)A = Ideg P1 ⊗ R(k), so
that R is a genuine representation of K . Similarly, for all l in L, we have
λ(l)A−1 D(l)A = P1 (l)⊗ν(l)Ideg P2 where ν is a map from L to C× ; defining
S(l) to be λ(l)−1 ν(l)P1 (l), we get A−1 D(l)A = S(l) ⊗ Ideg P2 , therefore S
FACTORIZATION OF REPRESENTATIONS
31
is a genuine representation of L. Now, for every element f in F , we can
choose k in K and l in L such that f = kl , obtaining
A−1 D(f )A = A−1 D(k)A A−1 D(l)A
= Ideg P1 ⊗ R(k) S(l) ⊗ Ideg P2
= S(l) ⊗ R(k)
= (S # R)(f ),
so that D is genuine-equivalent to R # S (swapping the factors does not
change the equivalence type), and both of R and S are irreducible. Finally,
recalling that σ is defined by σ(f ) := S̄(l), and observing that we have
S̄(l) = P̄1 (l) = P̄1 (f ), we conclude that σ = P¯1 ; in an entirely similar way
we also get % = P¯2 .
We are now in a position to prove the main results of the chapter.
2.8. Theorem. Let H be a group, and let F be a good group such that
F is a normal subgroup of H and CH (F ) is contained in F . Let D be
a faithful representation of H such that D ↓F is irreducible. Then there
exists a bijection between the set of all the pairs ([P1 ], [P2 ]), where P1 , P2
are projective representations of H such that D̄ ' P¯1 ⊗ P¯2 , and the set of
normal subgroups K of H such that K ≤ F and Z(K) = Z(H) hold. If
K corresponds to ([P1 ], [P2 ]) in this bijection, then we have K = ker(P¯1↓F )
and CF (K) = ker(P¯2↓F ).
Proof. We shall denote by Z the centre of H ; also, we shall denote
by FD the set of all the pairs ([P1 ], [P2 ]), where P1 , P2 are projective
representations of H such that D̄ ' P¯1 ⊗ P¯2 , and by S the set of normal
subgroups K of H such that K lies in F and Z(K) = Z . Next we observe
that, since D is faithful and its restriction to F is irreducible, we have
Z(F ) ≤ Z (this follows again by Schur’s Lemma); but F contains its own
centralizer in H , therefore Z(F ) coincides with Z .
Now, as the first step in the proof, we shall construct a map α from
FD to S : consider an element ([P1 ], [P2 ]) in FD and define α(([P1 ], [P2 ]))
as the kernel of P̄1 ↓F . Since equivalent projective representations yield
homomorphisms (to the relevant projective general linear group) which have
the same kernel, the ‘value’ α(([P1 ], [P2 ])) does not depend on the choice of
representatives for the classes [P1 ] and [P2 ]; moreover, denoting by K the
kernel of P̄1↓F , Lemma 2.7(d) tells us that Z(K) = Z , and certainly we have
32
2. ON TENSOR FACTORIZATION
K ≤ F and K H . The discussion above shows that α is actually a map
from FD to S . Also, again by Lemma 2.7(d), we get CF (K) = ker(P̄2↓F ).
As the second step we shall show that, given an element K of S , there
exists a unique element ([P1 ], [P2 ]) in FD such that ker(P̄1 ↓F ) = K ; this
will yield a map β from S to FD . So, let us start from an element K in
S ; by Lemma 2.7(a) we get F = KCF (K). If we denote by χ the character
afforded by D , we have χ↓F = ϕ for some ϕ in IrrF ; now K is a normal
subgroup of F and, if ϑ is an irreducible constituent of ϕ↓K , we clearly
have IF (ϑ) = F . We conclude that we have χ↓K = eϑ, where e is a positive
integer (this follows by Clifford’s Theorem (1.19)), hence we can assume
D↓K = Ie ⊗ T where T is an irreducible representation of K affording the
character ϑ. We are now in a position to apply Lemma 2.1, which ensures
the existence of an element ([P1 ], [P2 ]) in FD with the properties that K is
contained in ker(P̄1↓F ), deg P2 = deg T and T (xh ) = T (x)P2 (h) for all x in
K and h in H . We want now to prove that K coincides with ker(P̄1↓F ). Let
x be in ker(P¯1↓F ), and let k , c be elements, of K and CF (K) respectively,
such that x = kc (again we are using Lemma 2.7(a)). Since K is contained
in ker(P̄1 ↓F ), we have that c lies in ker(P̄1 ↓F ) as well, so that we have
D(c) = µIe ⊗ P2 (c) for some µ in C× ; moreover, c is in CF (K), hence
T (y)P2 (c) = T (y c ) = T (y) holds for all y in K , so that P2 (c) is a scalar
matrix, say νIdeg T for some ν in C× . We conclude that D(c) is given by
the Kronecker product of two scalar matrices, therefore c lies in Z (by the
faithfulness of D ) and our claim follows.
To complete the second step of the proof, we need to show that K
determines uniquely an element ([P1 ], [P2 ]) of FD with the property that
ker(P̄1↓F ) = K . For this purpose observe that, since we have D̄ ' P¯1 ⊗ P¯2 ,
there exist an element A of GL(deg D, C) and a map λ from H to C× such
that A−1 D(h)A = (λ(h))−1 P1 (h) ⊗ P2 (h) holds for all h in H . Moreover,
we get A−1 D(k)A = Ideg P1 ⊗ R(k) for all k in K , where R is the genuine
irreducible representation of K defined in the proof of Lemma 2.7(e). From
D(k h ) = D(k)D(h) we obtain now
Ideg P1 ⊗ R(k h ) = Ideg P1 ⊗ R(k)P2 (h) ,
whence R(k h ) = R(k)P2 (h) for all h in H and k in K . Since the genuineequivalence type of R is uniquely determined by K and by the genuineequivalence type of D , recalling Remark 2.2 we conclude that the projectiveequivalence type of P2 (that is, [P2 ]) is uniquely determined by K and by
the genuine-equivalence type of D . Similarly, we see that [P1 ] is uniquely
FACTORIZATION OF REPRESENTATIONS
33
determined by ker(P̄2 ↓F ) (which is in turn determined by K , being its
centralizer in F (Lemma 2.7(d))) and by the genuine-equivalence type of D .
It remains to observe now that α followed by β is the identity map
for FD , and β followed by α is the identity map for S ; this is completely
straightforward, and leads us to the end of the proof.
2.9. Definition. Let G be a group, and D a representation of G. We
define Z(D) and F (D) as the subgroups of G such that
Z(D)/ ker D = Z(G/ ker D)
and
F (D)/ ker D = F (G/ ker D).
2.10. Corollary. Let G be a group, and D a quasi-primitive representation
of G such that D↓F (D) is irreducible. There is a bijection between the set
of all the pairs ([P1 ], [P2 ]), where P1 and P2 are projective representations
of G such that D̄ ' P¯1 ⊗ P¯2 , and the interval [Z(D), F (D)] in the lattice
of normal subgroups of G. If ([P1 ], [P2 ]) corresponds to K in this bijection,
then we have
ker(P̄1↓F (D) ) = K
and
ker(P̄2↓F (D) ) = {x ∈ F (D) : [x, K] ⊆ ker D}.
Proof. Denoting by X the kernel of D , we consider the quotient group
Ĝ := G/X ; if ∆ is the representation of Ĝ defined by ∆(Xg) := D(g)
for all Xg in Ĝ, we have that ∆ is faithful and ∆ ↓F (Ĝ) is irreducible,
therefore CĜ (F (Ĝ)) is in the centre of Ĝ and, in particular, it lies in F (Ĝ).
This implies Z(Ĝ) = Z(F (Ĝ)) and, since ∆ is quasi-primitive, we conclude
that F (Ĝ) is a good group (see 2.6), which is obviously normal in Ĝ. Now
we are in a position to apply Theorem 2.8, obtaining that there exists a
bijection between the set F∆ of all the pairs ([Q1 ], [Q2 ]), where Q1 and Q2
¯ ' Q̄1 ⊗ Q̄2 , and the set of
are projective representations of Ĝ such that ∆
normal subgroups K̂ of Ĝ such that K̂ ≤ F (Ĝ) andZ(K̂) = Z(Ĝ). We
also know that, if ([Q1 ], [Q2 ]) corresponds to K̂ in this bijection, then we
have K̂ = ker(Q̄1↓F (Ĝ) ) and CF (Ĝ) (K̂) = ker(Q̄2↓F (Ĝ) ).
Consider now a projective representation P of G such that X lies
in ker P̄ ; we can choose a projective representation Q of Ĝ such that
Q̄(Xg) := P̄ (g) for all Xg in Ĝ and, associating [P ] with [Q], we can easily
construct a bijection between F∆ and the set of all the pairs ([P1 ], [P2 ]),
where P1 and P2 are projective representations of G such that D̄ ' P¯1 ⊗ P¯2 .
34
2. ON TENSOR FACTORIZATION
Also, the natural correspondence between normal subgroups of Ĝ and normal subgroups of G containing X provides, by restriction, a bijection between the set of normal subgroups K̂ of Ĝ such that K̂ ≤ F (Ĝ) and
Z(K̂) = Z(Ĝ), and the interval [Z(D), F (D)] in the lattice of normal subgroups of G; the proof can be now easily completed.
Factorization of characters
We give now an interpretation of the discussion above in terms of characters.
2.11. Definition. Let G be a group; we denote by (G̃, π) a Schur covering
of G (see 1.45 and the subsequent comment), and by A the kernel of π ,
which is a central subgroup of G̃; if H is a subgroup of G, we define H̃ as
π −1 (H).
If χ and ψ are irreducible characters of G, we say that they are equivalent (and we write χ ' ψ ) if there exists λ in IrrG such that λ(1) = 1
and χ = λψ . It is clear that, in this way, an equivalence relation on the set
IrrG is defined; we shall denote by [χ] the equivalence class of the character
χ modulo this equivalence relation.
Finally, we define Z(χ) and F (χ) in analogy with Definition 2.9 (the
notation Z(χ) is well established in the literature with a slightly different
meaning (see [Is], 2.26), but our definition is consistent with the usual one as
soon as χ is irreducible); observe that, if χinf is the character of G̃ obtained
from χ by inflation (see 1.35), we have F]
(χ) = F (χinf ).
2.12. Theorem. Let G be a group, and χ a quasi-primitive character of
G such that χ↓F (χ) is irreducible. Then the following properties hold:
(a) if N is a normal subgroup of G with Z(χ) ≤ N ≤ F (χ), then there exist
characters %1 and %2 of G̃ such that χinf ' %1 %2 and Z(%1↓F]
) = Ñ ;
(χ)
(b) let %1 , %2 , %3 and %4 be irreducible characters of G such that χ ' %1 %2
and χ ' %3 %4 ; if Z(%1 ↓F (χ) ) is the same as Z(%3 ↓F (χ) ), then we have
%1 ' %3 and %2 ' %4 ;
(c) there is a bijection between the set of all the pairs ([%1 ], [%2 ]), where
%1 and %2 are characters of G̃ such that χinf ' %1 %2 , and the interval [Z(χ), F (χ)] in the lattice of normal subgroups of G. If ([%1 ], [%2 ])
corresponds to N in this bijection, then we have Z(%1↓F]
) = Ñ .
(χ)
Proof of (a). Let D be a representation of G which affords χ; D is
quasi-primitive, its restriction to F (D) = F (χ) is irreducible, and N is
FACTORIZATION OF CHARACTERS
35
a normal subgroup of G with Z(D) ≤ N ≤ F (D); hence Corollary 2.10
yields that there exist projective representations P1 and P2 of G such that
D̄ ' P̄1 ⊗ P̄2 and ker(P̄1↓F (D) ) = N . By Theorem 1.45 we can find genuine
representations D1 and D2 of G̃, together with maps ξ1 and ξ2 from G̃
to C× , such that ξ1 (x)D1 (x) = P1 (Ax) and ξ2 (x)D2 (x) = P2 (Ax) for all
x in G̃ (here we are identifying G with G̃/A); it is now easy to see that
D , viewed by inflation as a representation of G̃, is projective-equivalent to
D1 ⊗ D2 . We conclude that, denoting by %1 and %2 the characters of G̃
afforded by D1 and D2 , we get χinf ' %1 %2 (see Remark 1.42); moreover,
it is easily checked that Z(%1↓F]
) coincides with Ñ .
(χ)
Proof of (b). Let Di (i ∈ {1, 2, 3, 4}) be representations which afford
the %i ; we have D̄ ' D̄1 ⊗ D̄2 and D̄ ' D̄3 ⊗ D̄4 ; moreover,
ker(D̄1↓F (D) ) = Z(%1↓F (χ) ) = Z(%3↓F (χ) ) = ker(D̄3↓F (D) )
holds. By Corollary 2.10 we conclude that D̄1 ' D̄3 and D̄2 ' D̄4 , so that
the claim follows.
Proof of (c). This is an immediate consequence of the two previous
statements.
2.13. Remark. Let G be a group, and P , Q, R projective representations
of G such that P̄ ⊗ Q̄ ' P̄ ⊗ R̄ =: D̄ ; if D happens to be a genuine quasiprimitive representation of G whose restriction to F (D) is irreducible, then
it follows from Corollary 2.10 that Q and R are equivalent (and therefore
we have, under the right assuptions, a ‘cancellation law’).
Even this claim fails if we weaken the hypothesis of irreducibility for
D↓F (D) , assuming only -for instance- that the restriction of D to F ∗ (D)
is irreducible (here F ∗ (D) denotes the preimage, under the natural homomorphism, of the generalized Fitting subgroup of G/ ker D ). Consider for
example G = A9 ; if we denote by P the 8-dimensional irreducible representation of G, by Q and R the two 21-dimensional irreducible representations
of G (which are inequivalent), and by D the 168-dimensional irreducible representation of G, we see that D is quasi-primitive (indeed primitive) and
of course irreducible when restricted to F ∗ (G). Moreover, D is equivalent
as a genuine representation to both of P ⊗ Q and P ⊗ R, and therefore we
have D̄ ' P̄ ⊗ Q̄ ' P̄ ⊗ R̄; but it is clear that Q and R are not equivalent,
even in a projective sense.
CHAPTER 3
On tensor induction
Two conjectures
In the discussion of Chapter 2 we have seen that, given a quasi-primitive
representation D for a group G, it is possible to parametrize, in terms of
the group structure of G, all the possible ways of factorizing D ; namely, we
have to look at normal subgroups of G which lie between Z(D) and F (D)
(recall Definition 2.9). The analysis of quasi-primitive representations gets
certainly a reduction by such a process, but at this stage it is natural to ask
what can be done if the relevant representation does not factorize at all, in
the sense of the following
3.1. Definition. Let G be a group, and D a representation of G. Then
D is called tensor-indecomposable if there do not exist two projective representations P1 and P2 of G, whose degrees are greater than 1, such that
D̄ ' P¯1 ⊗ P¯2 .
In the definition above we are again using notation from the last section of Chapter 1. If we are dealing with a quasi-primitive and tensorindecomposable representation D (which we also assume to be faithful, for
the sake of simplicity), a sensible way to reduce our analysis further is to
exploit the method of tensor induction for projective representations. More
precisely we ask if, given a subgroup H of G, it is possible to decide whether
D is tensor-induced (in the sense of Definition 1.39) by a projective representation of H , eventually recognizing such a projective representation. In
other words, we want to decide whether there exists (and how we might
recognize) a projective representation P of H such that D is projectiveequivalent to P̄↑⊗G . Our main problem (and our main task) for all the rest
of this work will be to give an answer to this question.
Let us assume the setting described above: D is a faithful, quasiprimitive and tensor-indecomposable representation of G, and we also assume that the Fitting subgroup F of G is not contained in the centre of
37
38
3. ON TENSOR INDUCTION
G, which we denote by Z (as already mentioned, this condition is certainly satisfied if G is a nonabelian solvable group). As we shall see in
the proof of Lemma 3.11, such a situation implies that the quotient group
F/Z is abelian of squarefree exponent, but the tensor indecomposability of
D ensures (through Corollary 2.10) that it is also a G-chief factor -there
aren’t normal subgroups of G strictly lying between Z and F - so that it
is a p-group for some prime number p, and it can be viewed as a simple
GF (p)[G]-module (with respect to the action of G given by conjugation).
Moreover, in 3.11(a) we see that F/Z is endowed (as a module) with a
particular G-invariant nonsingular symplectic form (see Definitions 3.4(c)
and 3.6), and the additive structure of this symplectic module turns out to
be very deeply linked to our main problem. Namely, Theorem 3.13 (which
takes advantage from the ‘Tensor Induction Theorem’ of L. G. Kovács (see
[Kov], sec. 6)) will show that, given a subgroup H of G, there exists an
‘explicit’ bijection between the set of (equivalence classes of) projective representations P of H such that D̄ ' P̄ ↑⊗G , and the set of submodules of
(F/Z)↓H which form-induce F/Z (as we already mentioned, form induction
is a kind of ordinary induction which ‘respects’ the structure given by the
symplectic form; see Definition 3.9). Thus, we can conclude that a problem
concerning the multiplicative structure of representations is translated into
a more comfortable additive problem, which after all comes from the group
structure of G (of course, we shall have to face the troubles which arise from
replacing the field C with a prime field); also, following the line of Chapter 2,
we can parametrize all the possible ways of tensor-inducing D from H in
terms of subgroups, lying between Z and F , which are normalized by H .
The analogy between ordinary induction and tensor induction suggests
another way in which the latter could be characterized: from 1.18(a) it
follows that, given a subgroup M of G, an irreducible representation S of
G, and a representation T of M , then S is induced by T from M if and
only if deg S = |G : M | deg T holds, and T is a ‘direct summand’ (in this
case, as irreducible constituent) of S↓M . Therefore, restricting to a solvable
context, we can formulate the following
3.2. Conjecture. Let G be a solvable group, H a subgroup of G, and D
a faithful, primitive, tensor-indecomposable representation of G. Assume
that we have D̄↓H ' P¯1 ⊗ P¯2 , where P1 and P2 are projective representations
of H . If (deg P2 )|G:H| = deg D (that is, P2 is a tensor factor of D↓H having
the ‘right’ degree), then D is tensor-induced by P2 from H (strong version
TWO CONJECTURES
39
of the conjecture) or, at least, D is tensor-induced from H , not necessarily
by P2 (weak version of the conjecture).
Observe that the converse of the conjecture above, in the strong version,
is an easy consequence of the definitions (assuming the setting of 3.2, such a
converse sounds as follows: if D is tensor-induced by P2 from H , then there
exists a projective representation P1 of H such that we have D̄↓H ' P¯1 ⊗P¯2 );
moreover, also the weak version of the conjecture would provide a very
convenient test for tensor induction, since the results of Chapter 2 keep
under control1 all the possible factorizations of D↓H .
The results of the following section, in particular Theorem 3.13, will turn
out to be very useful for approaching Conjecture 3.2: indeed, we shall see in
Remark 3.15 that they enable us to change our point of view, focusing on a
different problem:
3.3. Conjecture. Let G be a solvable group, H a subgroup of G, F a
finite field, V a simple FG-module, and W a submodule of V ↓H such
that V ' W ↑G . Assume also that V carries a G-invariant nonsingular
symplectic F-form which does not vanish on W . Then V is form-induced
by W from H (strong version of the conjecture) or, at least, V is forminduced from H , not necessarily by W (weak version of the conjecture).
The discussion of this chapter will be sufficient to draw some negative
conclusions about the two conjectures stated above. In the last section we
shall present two examples: the first of them definitely disproves the strong
version of both the conjectures, even in a setting where the subgroup H is
normal of odd prime index, whereas the second example disproves also the
weak version of the conjectures, but in a situation in which the index of H
in G is even.
What will be left is to investigate the validity of 3.2 and 3.3 in their
weak formulations, assuming that H has odd index in G; drawing positive
conclusions in this direction will be matter for later chapters.
1Indeed, assume that D is a faithful, primitive, tensor-indecomposable representation
of the (solvable) group G ; if we want to check whether D is tensor induced from a subgroup
H , first of all we have a necessary condition: H must contain F (see Lemma 3.11(b)).
Now, assume that the weak version of Conjecture 3.2 holds: then it is easy to see, using
2.7(b) and 2.8 (and recalling that in such a situation F is a ‘good group’ (see 2.5)), that
D is tensor-induced from H if and only if there exists a subgrup K of F such that K is
|G:H|
2
normal in H , Z(K) = Z(F ) , and |F/K|
= (deg D) .
40
3. ON TENSOR INDUCTION
Tensor induction and form induction
As we shall be dealing with symplectic forms and symplectic modules,
we need to introduce some terminology.
3.4. Definition. Let F be a field, and V a (finite dimensional) vector space
over F.
(a) Let f be a map from V × V to F; we say that f is a bilinear F-form
on V if it is F-linear with respect to both of the variables. It is worth
stressing that here, unlike in part of the literature, unitary forms are
not regarded as bilinear forms.
(b) If f is a bilinear F-form on V , then the set
Rad(f ) := {u ∈ V : f (u, v) = 0 for all v in V }
is a subspace of V , which is called the radical of f . We say that f is
nonsingular if Rad(f ) is the zero space.
(c) A bilinear F-form f on V is called symmetric if, for all u, v in V ,
we have f (u, v) = f (v, u), whereas it is called symplectic if f (v, v) = 0
holds for all v in V .
(d) Let f be a bilinear F-form on V which is symmetric or symplectic, and
W a subspace of V ; then the set
W ⊥ := {u ∈ V : f (u, w) = 0 for all w in W }
is also a subspace of V , called the orthogonal space
respect to f ).
2
of W in V (with
It will be useful to recall a well known result from linear algebra.
3.5. Theorem. Let V be a vector space over a field F, and let f be a
nonsingular bilinear F-form on V which is symmetric or symplectic. Then,
for every subspace W of V , the relation dim W + dim W ⊥ = dim V holds.
In particular, f is nonsingular on W if and only if we have V = W ⊕ W ⊥ .
Now we start merging the concepts of module and bilinear form; after
Definition 3.6 and Remark 3.7, which are given in greater generality, we
shall focus on symplectic forms, leaving to Chapter 4 a deeper discussion on
the relationship between modules and generic bilinear forms.
2The definition of orthogonal space can be given for generic bilinear forms, but in
that case it is necessary to distinguish between left and right orthogonality (this should
be done, in principle, also in defining the radical of a bilinear form).
TENSOR INDUCTION AND FORM INDUCTION
41
3.6. Definition. Let G be a group, F a field, V an FG-module, and f a
bilinear F-form on V ; if we have
f (ug , v g ) = f (u, v)
for all u, v in V and g in G, then we say that the form f is G-invariant
(in this situation G acts on V as a group of ‘f -isometries’).
3.7. Remark. If V is an FG-module which is endowed with a G-invariant
bilinear F-form, then it is easily seen that Rad(f ) is a submodule of V ;
this implies that, if V is simple and f is not the zero map on V × V , then
f is nonsingular. Moreover, if f is symmetric or symplectic and W is a
submodule of V , then W ⊥ is also a submodule of V (observe that, in this
situation, Rad(f ) is the orthogonal space of V ).
3.8. Definition. Let G be a group, F a field, and V an FG-module; if
f is a G-invariant nonsingular symplectic F-form on V , then we say that
V is a symplectic FG-module (with respect to f ). A submodule W of V
is called isotropic if it is contained in W ⊥ , whereas it is called anisotropic
if the only isotropic submodule of W is the zero space. Suppose now that
V is the direct sum of pairwise orthogonal submodules W1 , . . . , Wk (in the
sense that Wr lies in Ws⊥ for all s 6= r ); then V is the orthogonal direct
sum of the Wj , and we write
V = W1 ⊥ . . . ⊥Wk .
(See also [Da], sec. 1 and 2).
Finally, we can define the concept of form induction.
3.9. Definition. Let G be a group, H a subgroup of G, F a field, V a
symplectic FG-module (with respect to a given form f ) which is simple,
and W a submodule of V ↓H . Assume that the following conditions hold:
(a) the restriction of f to W × W , which is an H -invariant symplectic
F-form on W , is nonsingular;
(b) the translate W g lies in W ⊥ , for all g in G such that W g 6= W ;
(c) V is induced by W from H .
Then we say that V is form-induced by W (with respect to the form f )
from H .
42
3. ON TENSOR INDUCTION
We state here part of the ‘Tensor Induction Theorem’ in [Kov], Section 6 (recall the definition of tensor-indecomposability given in 3.1); the
subsequent lemma will provide, among other informations, a converse for it.
3.10. Theorem. Let G be a (not necessarily finite) group with a faithful, quasi-primitive and tensor-indecomposable representation D . Let L be
a noncentral subgroup which centralizes all its conjugates in G except perhaps itself, and set NG (L) = H . Then D is tensor-induced by a projective
representation P of H such that ker P̄ = CG (L).
3.11. Lemma. Let G be a group with noncentral Fitting subgroup F and
centre Z , and let D be a faithful, quasi-primitive, tensor-indecomposable
representation of G. Assume that H is a subgroup of G of index n,
{g1 = 1, . . . , gn } is a right transversal for H in G, and P is a projective
representation of H such that D̄ ' P̄ ↑⊗G . Then the following properties
hold.
(a) F/Z is a G-chief factor, therefore an elementary abelian p-group for
some prime p. As a GF (p)[G]-module, F/Z carries a G-invariant nonsingular symplectic form which comes from taking commutators in F .
(b) H contains F , so that we can define the subgroups K := ker(P̄↓F ) and
L := CF (K).
(c) If Q : F → GL((deg P )n−1 , C) is the projective representation of F
−1
N
defined by Q(y) := nj=2 P (y gj ) for all y in F , then we have
D̄↓F ' P̄↓F ⊗Q̄,
ker Q̄ = L
and
F = KL;
(d) F/Z is form-induced, as a GF (p)[G]-module, by L/Z from H , with
respect to the form of (a).
(e) P↓L is irreducible, and the projective-equivalence type of P is uniquely
determined by L and the genuine-equivalence type of D .
Proof of (a). First of all we observe that, D being quasi-primitive and
F being noncentral, the restriction of D to F is irreducible (otherwise,
by Lemma 2.1, D would not be tensor-indecomposable). Applying Schur’s
Lemma (1.3) we see that CG (F ) lies in Z , hence we get Z(F ) = Z and F
is a good group in the sense of 2.5 (again we use 2.6). Now, the assumption
of tensor indecomposability for D , together with Corollary 2.10, yields that
F/Z is a G-chief factor, hence a simple GF (p)[G]-module for some prime
p; moreover, since F/Z is an elementary abelian p-group, we have that F 0
TENSOR INDUCTION AND FORM INDUCTION
43
lies in Z and it has order p; therefore we can choose a generator (say x)
for F 0 , and define a map α : F × F → GF (p) by the following relations:
[y1 , y2 ] = xα(y1 ,y2 )
for all y1 , y2 in F .
If we define now a map f : F/Z × F/Z → GF (p) by setting
f (Zy1 , Zy2 ) := α(y1 , y2 ),
then it is straightforward to check that f is a G-invariant nonsingular symplectic form on F/Z .
Proof of (b). As a general fact, we observe that the kernel of a homomorphism of the kind P̄ ↑⊗G (P being a projective representation of H )
is the normal core of ker P̄ in G (if z is in ker(P̄ ↑⊗G ), which means that
(P↑⊗G )(z) is a scalar matrix, then z permutes trivially the right H -cosets in
its action by right multiplication; hence z lies in coreG (H), and (P↑⊗G )(z)
−1
−1
Nn
gj
is given by
). Now P (z gj ) is forced to be a scalar matrix
j=1 P (z
for all j in {1, . . . , n}). In our context, this implies in particular that
H contains Z . Consider now a minimal normal subgroup N/Z of G/Z ;
since F/Z is also minimal normal in G/Z , we have [N, F ] ≤ Z , so that
[N, F, N ] = [F, N, N ] = 1. Applying the ‘Three Subgroups Lemma’ (see
[Gor], 2.2.3), we get [N, N, F ] = 1, whence [N, N ] ≤ Z , and this means
that N/Z is abelian. By the discussion above, it is clear that F/Z is the
unique minimal normal subgroup in G/Z ; if H does not contain F , then
H/Z is core-free in G/Z , so that G/Z is embedded in the symmetric group
on n objects (which has order n!). Now (deg P )n = deg D is a divisor of
|G/Z| (recall that the degree of any irreducible representation of G is a
divisor of |G : Z|; see [Hu2], 6.5), but this is a contradiction, since n! is not
divisible by any n-th power. We conclude that claim (b) holds.
Proof of (c). This is clear, since F is contained in coreG H , and therefore
it acts trivially by right multiplication on the right H -cosets. The remaining
claims follow immediately from parts (c) and (d) of Lemma 2.7.
Proof of (d). We know that, for any element l of L, the matrix Q(l),
−1
−1
−1
which is given by P (lg2 ) ⊗ · · · ⊗ P (lgn ), is scalar; this forces Lgj
to be
−1
gj
contained in K for all j in {2, . . . , n}, hence we have [L , L] = 1 and
therefore [L, Lgj ] = 1 for all j in {2, . . . , n}. This means exactly that
(L/Z)g is contained in (L/Z)⊥ , with respect to the symplectic form defined
in (a), for all g with (L/Z)g 6= L/Z . Applying now Lemma 2.7(b), we get
|L/Z| = |F/K| = (deg P )2 , and also |F/Z| = (deg D)2 = (deg P )2n ; therefore L/Z is a submodule of (F/Z)↓H such that n dim(L/Z) = dim(F/Z).
44
3. ON TENSOR INDUCTION
This is sufficient to conclude that F/Z is induced by L/Z from H (see
1.18(a)) and, since it is clear that the relevant symplectic form restricted to
(L/Z) × (L/Z) is nonsingular on L/Z , the claim follows.
Proof of (e). Since the restriction of P to F is irreducible and F = KL,
where K is the kernel of P̄↓F , it is clear that P↓L is irreducible. Recalling
Definition 1.39, there exist an element A of GL(deg D, C) and a map λ
from G to C× such that λ(h)A−1 D(h)A = (P (h) ⊗ X(h))Y (h) for all h in
H , where X(h) is in GL((deg P )n−1 , C) and Y (h) is a permutation matrix
(on n objects) in GL(deg D, C) which fixes (acting by conjugation) the first
Kronecker factor; this is because h lies in StabG (H1) with respect to the
action of G on its right H -cosets by right multiplication. If l is an element of
L, then the equality above becomes A−1 D(l)A = S(l) ⊗ I(deg P )n−1 , where
S is the genuine irreducible representation of L defined in the proof of
Lemma 2.7(e). From D(lh ) = D(l)D(h) we obtain now
S(lh ) ⊗ I(deg P )n−1 = S(l)P (h) ⊗ I(deg P )n−1 ,
whence S(lh ) = S(l)P (h) for all h in H and l in L. Since the genuineequivalence type of S is uniquely determined by L and by the genuineequivalence type of D , recalling Remark 2.2, we get what we wanted.
After the next definition, we shall be in a position to prove the theorem
mentioned in the first section, which provides a link between the concepts
of tensor induction and form induction.
3.12. Definition. Let G be a group with Fitting subgroup F and centre
Z ; let H be a subgroup of G, and D a faithful, quasi-primitive, tensorindecomposable representation of G. We define the set
⊗G T ↑⊗G
.
H := [P ] : P a projective representation of H such that D̄ ' P̄ ↑
We also define the set
F ↑G
H := L : Z ≤ L ≤ F and F/Z is form-induced by L/Z from H
where, of course, form induction is meant with respect to the symplectic
form defined in Lemma 3.11(a).
3.13. Theorem. Let G be a group with noncentral Fitting subgroup F and
centre Z ; let H be a subgroup of G, and D a faithful, quasi-primitive,
tensor-indecomposable representation of G. There exists a bijection between
the set T ↑⊗G
and the set F ↑G
H . If [P ] corresponds to L in this bijection,
H
then we have L = CF (ker(P̄↓F )).
LINKING THE TWO CONJECTURES
45
Proof. Let L be an element of F ↑G
H ; then L is a noncentral subgroup of G which centralizes all its conjugates in G except perhaps itself,
and NG (L) = H . We can therefore apply the Tensor Induction Theorem
and conclude that there exists a projective representation P of H such
that D̄ ' P̄ ↑⊗G and ker(P̄ ↓F ) = CF (L). Since in this situation we have
F = L(ker P̄↓F ), L ≤ CF (ker(P̄↓F )) and ker(P̄↓F )∩CF (ker(P̄↓F )) = Z , we
conclude that L = CF (ker(P̄↓F )). Now, by Lemma 3.11(e), the projectiveequivalence type of P depends only on L and on the genuine-equivalence
⊗G
type of D , hence we can consistently define a map α from F↑G
H to T ↑H
G
setting α(L) := [P ]. Next, let β : T ↑⊗G
H → F ↑H be the map defined by
β([P ]) := CF (ker(P̄ ↓F )); from (b) and (d) of Lemma 3.11 it follows that
this is a good definition and, since it is clear that the β so defined is a
two-sided inverse to α, the result is proved.
Linking the two conjectures
In this section we shall see that, through Theorem 3.13, it is possible to
change our point of view in the analysis of Conjecture 3.2; more explicitly,
if we can prove Conjecture 3.3 (in the strong or in the weak version), then
the validity of Conjecture 3.2 (strong or weak, accordingly) follows as well.
The hypotheses of the next statement determine a situation, as in
Lemma 3.11, in which the quotient group F (G)/Z(G) (in the relevant group
G) is a G-chief factor; hence it is a p-group for a suitable prime number p,
and again we remark that it can be thought as a simple GF (p)[G]-module
with respect to the conjugation action of G. Moreover, by Lemma 3.11(a),
that module is endowed with a particular G-invariant nonsingular symplectic GF (p)-form. All these facts will be taken in account in stating and
proving Lemma 3.14.
3.14. Lemma. Let G be a solvable group with Fitting subgroup F and
centre Z , H a subgroup of G, and D a faithful, primitive, tensor-indecomposable representation of G. Assume that we have D̄↓H ' P¯1 ⊗ P¯2 , where
P1 and P2 are projective representations of H ; if deg P2 is not 1, and
(deg P2 )|G:H| is a divisor of deg D , then the following conclusions hold:
(a) the degree of D is equal to (deg P2 )|G:H| ;
(b) denoting by K the kernel of P̄1↓F , we have that K/Z is a submodule of
(F/Z)↓H , which induces F/Z from H . Moreover, the symplectic form
on F/Z defined in 3.11(a) is nonsingular on K/Z .
46
3. ON TENSOR INDUCTION
Proof. First of all, assume that the lemma holds for subgroups which
contain the centre Z , and consider the subgroup M := HZ . For any given
x in M , let h and z be elements, of H and Z respectively, such that
x = hz : we define the maps R̄i : M → P GL(deg Pi , C) (for i in {1, 2}) by
setting
R̄i (x) := P̄i (h)
for all x in M . This is certainly a good definition; indeed, if h1 z1 = h2 z2
for h1 , h2 in H and z1 , z2 in Z , then we get P̄i (h1 ) = P̄i (h2 ). Moreover, it is easily checked that R1 and R2 are projective representations of
M , and that D̄ ↓M ' R̄1 ⊗ R̄2 holds. Since (deg R2 )|G:M | is a divisor of
(deg R2 )|G:H| = (deg P2 )|G:H| , clearly (deg R2 )|G:M | is a divisor of deg D as
well; now the lemma gives us (deg R2 )|G:M | = deg D and this implies that
|G : M | = |G : H|, whence M = H . At this stage is clear that we can assume H ≥ Z and, as in Lemma 3.11(b), we see that H is forced to contain
the Fitting subgroup F of G.
Consider now the subgroup K := ker(P̄1↓F ): by Lemma 3.11(a), F/Z is
a G-chief factor, hence its order is a power of some prime p; recalling Lemma
2.7(b,c) we see that |K/Z| = (deg P2 )2 , therefore deg P2 = pr for a suitable
integer r , and |K/Z| = p2r ; similarly we have |F/Z| = (deg D)2 and, since
(deg P2 )|G:H| is a divisor of deg D , we have deg D = pr|G:H|+w for some
integer w , whence |F/Z| = p2(r|G:H|+w) . Clearly K/Z is a submodule of
Q
(F/Z)↓H and, if T is a transversal for NG (K) in G, we get F = g∈T K g
because F/Z is a simple GF (p)[G]-module. Looking at the dimensions
of K/Z and F/Z as GF (p)-vector spaces, we conclude that w = 0 and
h = |G : H| (this implies NG (K) = H and (deg P2 )|G:H| = deg D ); moreover, in a vector space notation we have
|G:H|
(F/Z)↓H =
M
(K/Z)gj ,
j=1
so that K/Z induces F/Z from H . Finally, it is easily seen that the relevant symplectic form on F/Z is nonsingular on K/Z ; indeed, we have
F = KCF (K) (here we use Lemma 2.7(c,d)), so that F/Z = K/Z ⊕(K/Z)⊥
holds and, by 3.5, the conclusion is achieved.
3.15. Remark. At this stage it is easy to see that, if Conjecture 3.3 is
proved, then 3.2 follows at once. Suppose we want to prove the weak version
of 3.2: first of all we go through Lemma 3.14, getting that the symplectic
simple GF (p)[G]-module F/Z is induced by K/Z (here of course K still
EXAMPLES
47
denotes the kernel of P¯1↓F ), which is an anisotropic submodule of (F/Z)↓H .
Now the weak version of 3.3 yields that F/Z is form-induced from H and,
by Theorem 3.13, we obtain immediately the desired conclusion.
Also, if the strong version of 3.3 were proved, we would have that F/Z is
form-induced from H by K/Z and, this time, Theorem 3.13 yields a projective representation P of H such that D̄ ' P̄↑⊗G and ker(P̄↓F ) = CF (K).
Now, we know that any projective representation which tensor-induces D
from H is a tensor factor of D↓H , that is, there exists a projective representation Y of H such that D̄↓H ' Ȳ ⊗ P̄ holds; but we also have D̄↓H ' P¯1 ⊗ P¯2
and, since the kernel of P¯2 ↓F is also CF (K), Theorem 2.8 ensures that P
and P2 are equivalent.
We shall see in a moment that the strong version of both our conjectures
fails; at any rate, the second part of the remark above is not useless, since
(assuming the hypothesis of Lemma 3.14) it emphasizes that the tensor
factor P2 does tensor-induce D as soon as (ker(P¯1 ↓F ))/Z form-induces
F/Z .
Examples
We shall go now through the discussion of two examples. The first of
them presents the construction of a solvable group G which has a faithful,
primitive, tensor-indecomposable representation D of degree 53 , and which
also has a normal subgroup H of index 3. It is possible to find projective
representations P1 and P2 of H such that D̄↓H ' P¯1 ⊗ P¯2 , and such that
deg P2 = 5; nevertheless, D is not tensor-induced by P2 from H (this is
essentially achieved after we have built a counterexample for the strong version of Conjecture 3.3). In other words, Conjecture 3.2 in the strong version
is disproved. This tells us that we can not pursue the analogy between ordinary induction and tensor induction as far as we could expect, even if the
subgroup involved is normal of odd prime index.
The second example presented below shows that both conjectures fail
even in their weak version, in a situation in which the subgroup H has even
index.
Before going through the examples, we recall a general result concerning
semisimple modules over finite fields, which will be useful in 3.18 and also
in the appendix of the thesis.
3.16. Lemma. Let G be a group, F a finite field, and U a semisimple
homogeneous FG-module whose composition length is m. Then, denoting
48
3. ON TENSOR INDUCTION
by V a simple submodule of U , and by q the order of EndFG (V ), the number
Pm−1 j
of simple submodules of U is given by
j=0 q .
(See [HB], VII, proof of 9.19).
3.17. Example. Consider the group
E := ha, b : a5 = b5 = [a, b]5 = 1, [a, b, a] = [a, b, b] = 1i
(E is the extraspecial 5-group of order 53 and exponent 5). In Aut(E) we
can find an element σ which maps a to b, and b to b−1 a−1 ; it is easy to
verify that σ has order 3, and that it centralizes E 0 . Consider now the
automorphism τ of E × E × E defined by (x, y, z)τ := (zσ, x, y); if E0
is the subgroup of E × E × E which consists of the elements (x, y, z) in
E 0 × E 0 × E 0 such that xyz = 1, we see that E0 is normalized by τ . We
conclude that τ is an automorphism, which has order 9, of the extraspecial
5-group F := (E × E × E)/E0 . If C9 is a 9-cycle generated by the element
t, it is now possible to form the semidirect product F oC9 setting y t := yτ
for all y in F ; G has order 32 · 57 , and it has a normal subgroup of index 3,
namely H := F oht3 i.
Next, we choose an irreducible character ϕ of F with ϕ(1) 6= 1; we have
that ϕ is faithful of degree 53 and, since τ centralizes Z(F ), the inertia subgroup of ϕ in G is the whole G (see [Hu2], 7.5). This, together with the fact
that G/F is cyclic, ensures ([Hu2], 22.3) that there exists an irreducible character χ of G whose restriction to F is ϕ. Clearly F is the Fitting subgroup
of G, whence χ is also faithful, and we have Z(F ) = Z(H) = Z(G); denoting Z(F ) by Z , we also observe that F/Z is a simple GF (5)[G]-module
(indeed, any submodule of F/Z should be invariant under the action of t,
which after restriction is still an automorphism of order 9; but no vector
space of dimension 1, 2 or 3 over GF (5) admits such an automorphism).
Finally we see that χ is primitive, because G does not have any proper subgroup whose index is a divisor of 53 (suppose that M is such a subgroup:
we can assume that M contains t, so that the unique 5-Sylow subgroup X
of M has to be normalized by t; now, XZ = XΦ(F ) would be a normal
subgroup of G properly contained in F (here Φ(F ) denotes the Frattini
subgroup of F ), which is not the case); at this stage Corollary 2.10 ensures that, denoting by D the representation of G which affords χ, D is
tensor-indecomposable.
Consider now the simple GF (5)[G]-module F/Z : as in Lemma 3.11(a),
we can define a G-invariant nonsingular symplectic form on it by taking
EXAMPLES
49
commutators in F . Denoting by v1 the right Z -coset of P0 (a, 1, 1), and by
v2 the right Z -coset of P0 (b, 1, 1), the subspace V := hv1 , v2 i is indeed an
2
anisotropic simple submodule of (F/Z)↓H , and we have F/Z = V ⊥V t ⊥V t .
2
2
With respect to the basis {v1 , v2 , v1t , v2t , v1t , v2t } the form is given by a blockscalar matrix, each block being a hyperbolic plane. If now W is the subspace
spanned by (0, −1, 0, −1, 0, −1) and (1, 1, 1, 1, 1, 1), it is clear that W is an
anisotropic simple submodule of (F/Z)↓H , such that W t is not contained
in W ⊥ ; in other words, W induces F/Z without form-inducing it (using
the results of Chapter 4 it can be shown that, among the 525 anisotropic
simple submodules of (F/Z)↓H , only 21 form-induce F/Z ).
We are now in a position to achieve the conclusion: let us denote by L
the subgroup of G such that L/Z = W ; clearly we have F = LCF (L),
hence Z(L) coincides with Z , and Theorem 2.8 tells us that there exist projective representations P1 and P2 of H such that D̄ ↓H ' P¯1 ⊗ P¯2
and L = ker(P¯1 ↓F ). Moreover, we have (deg P2 )2 = |L/Z|, hence P2
has degree 5; but if D were tensor-induced by P2 then, by Theorem 3.14,
L = CF (ker P¯2 ↓F ) should provide (by taking L/Z ) an H -submodule of
F/Z which form-induces F/Z . As we saw, this is definitely not the case.
3.18. Example. As in the previous example, we start from the extraspecial
5-group E := ha, b : a5 = b5 = [a, b]5 = 1, [a, b, a] = [a, b, b] = 1i of order 53
and exponent 5. In Aut(E) there exist an element i which maps a to b−1
and b to a, and an element j which maps a to a2 and b to b−2 ; it is easy to
check that i and j generate in Aut(E) a subgroup K which is isomorphic
to the quaternion group of order 8, and which centralizes E 0 . Let now C2
be a 2-cycle with generator x; denoting by Y the semidirect product EoK ,
we have that E × E is a normal subgroup of Y o C2 . Moreover, if E0 is the
subgroup of E × E consisting of the elements (q, r) in E 0 × E 0 such that
qr = 1, then E0 is also normal in Y o C2 . In particular, K o C2 acts as a
group of automorphisms on the extraspecial 5-group F := (E ×E)/E0 ; also,
it is straightforward that Z(F ) is centralized by K o C2 . Consider now the
elements t := (i, 1)x and s := (j, j)x in K o C2 ; the subgroup Q of K o C2
generated by t and s is isomorphic to the quaternion group of order 16. We
are finally in a position to define the group G := F oQ, and its subgroup
H := F oht2 , si (observe that ht2 , si is isomorphic to the quaternion group
of order 8).
Next, consider an irreducible character ϕ of F such that ϕ(1) 6= 1:
ϕ is faithful of degree 52 , its inertia subgroup in G is all of G and, since we
50
3. ON TENSOR INDUCTION
have (|F |, |G : F |) = 1, there exists an irreducible character χ of G whose
restriction to F is ϕ (see [Hu2], 22.3). As in Example 3.17, F is the Fitting
subgroup of G, hence χ is faithful and we have Z(F ) = Z(H) = Z(G).
Denoting by Z the centre of F , it is easy to check that F/Z is a simple
GF (5)[G]-module; moreover, G does not have any proper subgroup whose
index divides 52 , so that χ is primitive and the representation D of G
which affords χ is tensor-indecomposable.
The simple GF (5)[G]-module F/Z is endowed with the usual nonsingular G-invariant symplectic form which arises by taking commutators in
F ; if we choose, as a basis for F/Z , the right Z -cosets of P0 (a, 1), P0 (b, 1),
P0 (1, a) and P0 (1, b) (let us denote them by v1 , v2 , v3 and v4 ), we see that
the form is given by a block-scalar matrix having hyperbolic planes as blocks.
Moreover, t maps v1 to −v4 , v2 to v3 , v3 to v1 and v4 to v2 , whereas s
maps v1 to 2v3 , v2 to −2v4 , v3 to 2v1 and v4 to −2v2 . It is now easy
to check that the subspace W of F/Z spanned by (1, 0, 1, 0) and (0, 1, 0, 1)
is an anisotropic simple submodule of (F/Z)↓H , such that W t is not contained in W ⊥ . If (F/Z)↓H were inhomogeneous, its only proper nonzero
submodules would be the two homogeneous components, so we would be
forced to have W t = W ⊥ . Thus (F/Z)↓H is the direct sum W ⊕ W t of
two isomorphic simple submodules. Since | EndH (W )| = 5, (F/Z)↓H has
precisely six simple submodules (see Lemma 3.16). Among them, we find
two isotropic submodules (one spanned by (1, 2, 2, 1) and (2, −1, 1, −2), the
other the image of this under t). It is then clear that (W ⊥ )t = (W t )⊥ , hence
the pair (W ⊥ , (W ⊥ )t ) does not yield an orthogonal direct decomposition of
F/Z as well. In other words, F/Z is not form-induced by any submodule
of (F/Z)↓H , thus proving that Conjecture 3.3 fails if the index of H is not
assumed to be odd.
In order to show that an odd-index assumption is needed also for Conjecture 3.2, let us consider the subgroup L of G such that L/Z = W ;
exactly the same argument applied in Example 3.17 shows that there exist projective representations P1 and P2 of H such that D̄ ↓H ' P¯1 ⊗ P¯2
and L = ker(P¯1 ↓F ), where P1 and P2 have degree 5. Nevertheless, D
is not tensor-induced by any projective representation of H ; otherwise, by
Theorem 3.13, F/Z would be form-induced from H .
CHAPTER 4
Modules and bilinear forms
A further reduction
In the previous chapter a conjecture on tensor induction for complex
representations was presented, and a first approach to it led us to a rather
different area of Representation Theory; namely, we got a reduction of the
original conjecture to a problem (Conjecture 3.3) which concerns the additive structure of symplectic modules over finite fields. Now, following this
line, it will be convenient to carry out a deeper analysis on the interaction
between modules and bilinear forms.
Let V be a simple FG-module, where F is a finite field. In the paragraphs numbered from 4.3 to 4.6 we shall see that, as soon as a G-invariant
nonsingular bilinear F-form f is defined on V (recall 3.4 and 3.6), a bijection
between the set B := {G-invariant nonsingular bilinear F-forms on V }
and AutFG (V ) can be established; also, through the form f it is possible to
define a field automorphism on EndFG (V ), denoted by τ , whose order turns
out to be at most 2. While the bijection mentioned above depends actually
on the form f , the automorphism τ does not change if it is constructed from
another element of B . Indeed, Theorem 4.12 and Theorem 4.13 show that τ
depends only on the module structure of V (in particular, on the structure
of V regarded in a natural way as a module over the field EndFG (V )).
As a first relevant fact concerning the automorphism τ , we see that it
provides some useful information about the ‘nature’ of a G-invariant nonsingular bilinear F-form on V (in the sense that, once we have chosen a form in
B which is -for instance- symmetric, by means of τ we can decide whether
another given form in B is also symmetric or not; see paragraph 4.7). This
will enable us to characterize, in terms of τ , which types of forms (or, at
least, how many types of forms) appear in the set B ; without going into
many details here (see the discussion after Theorem 4.13) we have that two
different situations may occur, provided B is not an empty set, depending
on whether τ is the identity or not.
51
52
4. MODULES AND BILINEAR FORMS
Another context in which the automorphism τ plays a central role,
especially for our purposes, is the following: assuming that the set B contains, say, a symplectic form (but nothing changes in the symmetric case),
it is possible to define a convenient equivalence relation on the subset S
of B constituted by the symplectic forms (see paragraph 4.14). Modulo
this equivalence relation, the set S is partitioned into at most two equivalence classes, and again we have a full control of the situation in terms of
τ ; indeed, there are two distinct equivalence classes if and only if τ is the
identity (unless F has characteristic 2; in that case we always have a unique
equivalence class (see Proposition 4.16)).
The last critical ingredient, in order to approach Conjecture 3.3 along
this line, is the concept of induced form; assume that H is a subgroup of G,
and W is a submodule of V ↓H which carries an H -invariant nonsingular
symplectic F-form g . If W induces V from H , then it is possible to construct, from g , a G-invariant nonsingular symplectic F-form on V , which
we denote by g↑V ; this is achieved by the process of ‘induction of forms’
defined in paragraph 4.17.
We are now in a position to sketch how all this machinery can be exploited in our context; in paragraph 4.20 we draw the following conclusion.
Let G be a group, H a subgroup of G, F a finite field, V a simple
FG-module, and W a submodule of V ↓H . Assume that W induces V from
H , and that W carries an H -invariant nonsingular symplectic F-form.
Then, for any G-invariant nonsingular symplectic F-form h , i on V , the
following conditions are equivalent:
(a) the form h , i is equivalent to a form which is induced from W ,
(b) V is form-induced, with respect to h , i, by a submodule of V ↓H isomorphic to W .
With such a characterization it is not hard to guess (at any rate, it will
be clear going through the chapter) that Conjecture 3.3 is certainly true
if there is a unique equivalence class of G-invariant nonsingular symplectic
F-forms on V ; as we mentioned, this happens exactly when τ is not the
identity on EndFG (V ) if F has odd characteristic, but it always happens
if F has characteristic 2. Hence we can focus on a situation in which F
has odd characteristic and τ is the identity. But Lemma 4.22 shows that, in
this setting, there exists a G-invariant nonsingular symplectic F-form on V
which is not equivalent to a form obtained by induction from W if and only if
| EndFG (V ) : EndFH (W )| is an even number (here | EndFG (V ) : EndFH (W )|
BILINEAR FORMS ON A SIMPLE MODULE
53
denotes the degree of EndFG (V ) as a field extension of EndFH (W ); we shall
see in paragraph 4.18 that the second field can be embedded in the first
one).
In conclusion, we can say that the weak version of Conjecture 3.3 (with
the extra assumption1 that the index of H in G is odd) holds if we can
prove the following
4.1. Conjecture. Let G be a solvable group, H a subgroup of G having
odd index, F a finite field of odd characteristic, V a simple FG-module, and
W a submodule of V ↓H such that V ' W↑G . Assume also that W carries
an H -invariant nonsingular symplectic F-form. Then either τ is not the
identity on EndFG (V ), or there exists a submodule Z of V ↓H , which also
carries an H -invariant nonsingular symplectic F-form, such that V ' Z↑G
and | EndFG (V ) : EndFH (Z)| is an odd number.
Such a statement appears to be rather technical, but we shall see in
the following chapter that it provides a convenient reduction for the original problem; observe that 4.1 would give us more than we need (at least
‘a priori’), since it implies that all the G-invariant nonsingular symplectic
F-forms on V are equivalent to a form which is induced from the same
submodule Z , whereas we are interested in proving that each of them is
equivalent to an induced form, but not necessarily from the same Z .
This section was meant to give an idea of the process leading to this
reduction; the rest of the chapter is devoted to actually achieve it.
Bilinear forms on a simple module
Before going through the discussion, it is worth adding a comment to
Definition 3.4: if V is a vector space over a field F of odd characteristic,
then a bilinear F-form f is symplectic if and only if f (v, w) = −f (w, v)
holds for all v , w in V , and of course the symmetric and the symplectic
are two different types of forms; but if F has characteristic 2, then we see
that a symplectic form is just a particular kind of symmetric form. Also, we
recall the concept of contragredient module.
4.2. Definition. Let G be a group, F a field, and V an FG-module.
Viewing V as an F-vector space, its dual space V ∗ is the set of all the
1Recall that, if such an hypothesis is dropped, Conjecture 3.2 and Conjecture 3.3 are
definitely false (see 3.18).
54
4. MODULES AND BILINEAR FORMS
F-linear maps from V to F, endowed with a structure of F-vector space
given by
v(αλ) := (vα)λ
for all v in V , λ in F, and α in V ∗ . It is possible to define an action of G
on V ∗ , setting
−1
v(αg ) := (v g )α
for all v in V , g in G, and α in V ∗ , and in this way V ∗ acquires a structure
of FG-module. This module is denoted by V ∗ as well, and it is called the
contragredient module of V ; of course V is simple if and only if V ∗ is so,
and it can be easily checked that, if T is a representation associated to V ,
then V ∗ is associated to the representation which maps every element g in
G to the transpose-inverse matrix of T (g). Finally, if V ∗ is isomorphic to
V as an FG-module, then we say that the module V is self-contragredient.
4.3. In this paragraph we assume that F is a field, V is a finite dimensional
vector space over F, and f : V × V → F is a nonsingular bilinear form.
First of all we observe that the map ψf , from V to the dual space V ∗ ,
defined by u(vψf ) := f (u, v), is an isomorphism of vector spaces. Consider
now an element ε in EndF (V ); given a v in V , the map
ε(vψf ) : V → F, u 7→ f (uε, v)
is in V ∗ . Therefore, there exists a unique v 0 in V such that ε(vψf ) = v 0 ψf ,
that is f (uε, v) = f (u, v 0 ) for all u in V . In this way a new map ε0 can
be defined by setting vε0 := v 0 for all v in V , and we see that ε0 is an
element of EndF (V ). Moreover, the map τf which associates ε to ε0 turns
out to be an antiautomorphism of the algebra EndF (V ). Next, let g be
another nonsingular bilinear F-form on V ; for any given v in V the map
vψg is in V ∗ , hence there exists a unique v 0 in V such that vψg = v 0 ψf ,
which means g(u, v) = f (u, v 0 ) for all u in V . If we define a map γ by
setting vγ := v 0 for all v in V , we see that γ is in EndF (V ) but, since g
is nonsingular, γ is actually an element of AutF (V ). Conversely, chosen γ
in AutF (V ), the map g from V × V to F defined by g(u, v) := f (u, vγ) is
a nonsingular bilinear form. In conclusion, we see that a bijection between
the set of nonsingular bilinear F-forms on V and AutF (V ) arises by means
of the given form f . Also, if the form g corresponds to γ in this bijection,
and if τg is the antiautomorphism of EndF (V ) attached to g , then it is easy
to check that the maps τg and τf are linked by the relation τg = τf Inn(γ −1 )
where Inn(γ −1 ) : EndF (V ) → EndF (V ) maps ε to γεγ −1 .
BILINEAR FORMS ON A SIMPLE MODULE
55
4.4. We move now to a situation in which a group G is also playing;
more precisely, we assume here that G is a group, F is a field, V is an
FG-module, and f : V × V → F is a nonsingular bilinear form which is
also G-invariant. In this richer context, the vector space isomorphism ψf
defined in 4.3 turns out to be an isomorphism of FG-modules. Moreover,
consider the correspondence between the set of nonsingular bilinear F-forms
on V and AutF (V ) which is determined by f as in 4.3; it is easy to verify that the subset of nonsingular bilinear F-forms on V which are also
G-invariant is now bijective with AutFG (V ).
4.5. As we have seen in 4.3, if f is a nonsingular bilinear F-form on the
F-vector space V , then f defines a vector space isomorphism from V to
its dual space. It is worth recalling that also the converse is true; more
precisely, let ψ be an isomorphism of vector spaces from V to V ∗ : then
a nonsingular bilinear F-form on V , which we denote by fψ , arises in a
natural way setting fψ (u, v) := u(vψ) for all u, v in V . This can be also
conveniently specialized to the context of 4.4; in fact, if ψ : V → V ∗ is an
isomorphism of FG-modules, then the form fψ defined above turns out to
be G-invariant. In particular, there exist G-invariant nonsingular bilinear
F-forms on V if and only if V is self-contragredient.
4.6. Let us now specialize the setting further, assuming that G is a group,
F is a finite field, V is a simple FG-module, and f : V × V → F is a
G-invariant nonsingular bilinear form. In this case EndFG (V ) is a field
(see 1.3), and the restriction to it of the map τf defined in 4.3 turns out to
be a field automorphism. Moreover, if we start from another G-invariant
nonsingular bilinear F-form on V , say g , of course we have that τg agrees
with τf on EndFG (V ) (see the last sentence in 4.3 and in 4.4). Therefore,
in the present context, we are allowed to denote this field automorphism
simply by τ , dropping any reference to a distinguished form. At any rate,
it will be convenient sometimes to relate explicitly τ to the module V ; in
that case, we use the notation τV .
4.7. Let us assume the same setting of 4.6 with the additional hypothesis
that the characteristic of F is odd, and let the form f be symmetric or
symplectic. It is easy to check that, if g is another G-invariant nonsingular
bilinear F-form on V , then g is of the same type as f (that is, symmetric or
symplectic) if and only if the element γ of EndFG (V ) which is associated to
g (in the bijection defined by f ) is such that γ τ = γ . Assume now that the
56
4. MODULES AND BILINEAR FORMS
field F has characteristic 2; in this case it still holds that, if f is symmetric,
then g is also symmetric if and only if γ is fixed by τ .
We are now in a position to prove two easy lemmas. It will be useful to
have a name for a bilinear form which in neither symmetric nor symplectic;
we shall call such a form elusive.
4.8. Lemma. Let G be a group, F a field, V a simple FG-module, and
f a G-invariant nonsingular bilinear F-form on V . If f is elusive, then
a G-invariant nonsingular symmetric F-form can be defined on V and,
provided the characteristic of F is not 2, V carries also a G-invariant
nonsingular symplectic F-form.
Proof. Let us assume that the characteristic of F is not 2, and let us define the maps s and a, from V ×V to F, by setting s(u, v) := f (u, v)+f (v, u)
and a(u, v) := f (u, v) − f (v, u). It is straightforward that both s and a are
G-invariant bilinear F-forms, which are respectively symmetric and symplectic. Assume now that s is singular; since V is a simple FG-module,
then s is indeed zero (recall 3.7). But this implies that f is symplectic, a
contradiction. Similarly, if a is assumed singular, we get that f is symmetric, again a contradiction. We conclude that both s and a are nonsingular,
as desired.
If the characteristic of F is 2, then clearly the two maps s and a turn out
to be the same G-invariant nonsingular symmetric form. As above, if this
form were singular then f would be symmetric, against the assumption.
We note here that the previous lemma implies that, as soon as a simple
FG-module is self-contragredient, it admits a symmetric or a symplectic
G-invariant nonsingular F-form; this turns to be critical for the next lemma.
It is also worth mentioning that, if F is a finite field of characteristic 2, any
such module -except the trivial one- carries indeed a G-invariant nonsingular
symplectic F-form (see [HB], VII, 8.13).
4.9. Lemma. Let G be a group, F a finite field, and V a simple FG-module
which carries G-invariant nonsingular bilinear F-forms. Then the order of
the field automorphism τ on EndFG (V ) is at most 2.
Proof. Let f be a G-invariant nonsingular F-form on V which is
symmetric or symplectic; the existence of such a form is guaranteed by
BILINEAR FORMS ON A SIMPLE MODULE
57
Lemma 4.8, as we observed. Now, if n takes value 1 or −1 in F accordingly to the symmetric or symplectic nature of f , we have
2
f (x, yετ ) = f (xετ , y) = nf (y, xετ ) = nf (yε, x) = f (x, yε)
for all x, y in V and ε in EndFG (V ). The claim follows now from the fact
that f is nonsingular.
Before proving Theorem 4.12, which is useful for understanding the role
played by the automorphism τ , some preparation is needed; if G is a group,
F a finite field, and V a simple FG-module, then E will denote the field
EndFG (V ), and VE the simple EG-module which arises regarding V as
a vector space over E (we consider here the natural action of E on V ),
whereas V E will be the EG-module defined in 1.5. Also, we introduce the
following
4.10. Definition. Let G be a group, F a field, K a Galois extension of F,
and U a KG-module. If η is an element of the Galois group Gal(K|F), then
we denote by U η the KG-module which arises as follows: the underlying
set is the same as for U , and the action of G is unchanged as well, but the
action of K is ‘twisted’ by η , being defined by
uε := uεη
−1
for all u in U η (in the right-hand side u is treated as an element of U ,
with K acting on it in the original fashion). See [HB], VII, 1.13 for further
details.
The next lemma provides a link between the concepts introduced above;
it is worth recalling here that any finite extension of a (finite) field F is
certainly a Galois extension.
4.11. Lemma. Let G be a group, F a finite field, V a simple FG-module,
and E the endomorphism ring of V . Then we have
M
VE '
(VE )η ,
η∈Gal(E|F)
where the direct summands are pairwise nonisomorphic simple EG-modules.
Proof. This follows immediately from [HB], VII, 1.15 and 1.16a).
4.12. Theorem. Let G be a group, F a finite field, V a simple FG-module,
and f a G-invariant nonsingular bilinear F-form on V . Then f yields an
isomorphism of EG-modules between (VE )τ and (VE )∗ .
58
4. MODULES AND BILINEAR FORMS
Proof. Consider the F-vector space V ∗ ; it is possible to define an action
of E on V ∗ setting vδ ε := (vε)δ for all δ in V ∗ and ε in E; in this way
we obtain an E-vector space which we denote by (V ∗ )E (this notation is
consistent with the convention established above, since EndFG (V ∗ ) is E,
and the natural action of it on V ∗ is exactly the one we just described).
Next, recall that (VE )τ is an E-vector space (indeed, an EG-module) which
is the same as VE as a set, but the action of E is twisted by τ , so that v ε
−1
is set to be vετ (that is vετ , since τ has order at most 2) for all v in VE
and ε in E.
Consider now the map ψf defined in 4.3; we know that ψf is an
FG-isomorphism from V to V ∗ . We claim that it provides also an
EG-isomorphism from (VE )τ and (V ∗ )E ; indeed, for all x, v in V and
ε in E we get
x(v ε ψf ) = f (x, v ε ) = f (x, vετ ) = f (xε, v) = (xε)(vψf ) = x(vψf )ε ,
hence ψf is an isomorphism of E-vector spaces. Moreover, recalling that the
−1
relevant structure of EG-module on (V ∗ )E is defined by x(δ g ) := (xg )δ
for all x in V and δ in (V ∗ )E , it is easily checked that ψf is actually an
isomorphism of EG-modules.
The final step consists in showing that (V ∗ )E is isomorphic, as an
EG-module, to (VE )∗ . For this purpose, we choose a nonzero F-linear
map µ from E to F, and we define the map β : (VE )∗ → (V ∗ )E by
v(ϕβ) := (vϕ)µ for all v in V and ϕ in (VE )∗ . It is routine to check
that β is an EG-homomorphism. Also, if ϕ is a nonzero element of (VE )∗ ,
then its image is E; now the image of ϕβ is F, so that ϕβ is not zero as
well. This shows that β is actually an isomorphism, which completes the
proof.
The next theorem shows that the behaviour of the automorphism τ is
deeply linked to the structure of the module V E , but before stating it we
need the following observation: let f be a bilinear form on V , and let A
be the matrix associated to f with respect to a given F-basis {v1 , . . . , vn }.
Now, {v1 ⊗ 1, . . . , vn ⊗ 1} is an E-basis for V E and A can be regarded as
a matrix with entries in E, to which we can associate a bilinear E-form f¯.
We refer to f¯ as to the E-linear extension of f ; of course, properties like
G-invariance or nonsingularity are inherited by f¯ if they hold for f .
4.13. Theorem. Let G be a group, F a finite field, V a simple FG-module,
and f a G-invariant nonsingular bilinear F-form on V . Let f¯ be the
BILINEAR FORMS ON A SIMPLE MODULE
59
E-form on V E which arises as the E-linear extension of f . Then the following conditions are equivalent:
(a) the automorphism τ is the identity on E,
(b) the module VE is self-contragredient,
(c) the form f¯ does not vanish on any simple submodule of V E , and any
two distinct simple submodules are orthogonal with respect to f¯.
Also, the other possible picture is described by the following equivalent conditions:
(a 0 ) the automorphism τ is not the identity on E,
(b 0 ) the module VE is not self-contragredient,
(c 0 ) the module V E has a direct decomposition such that the restriction of f¯
to any direct summand is nonsingular; the direct summands are pairwise orthogonal with respect to f¯, and each of them is the direct sum
of two simple submodules on which f¯ vanishes and which are contragredients of each other.
Proof. Recall that, by 4.11, we have
M
VE '
(VE )η
η∈Gal(E|F)
where the direct summands are pairwise nonisomorphic simple EG-modules.
Theorem 4.12 proves that (a) implies (b). Since τ is an element of
Gal(E|F) and the Galois-conjugates of VE are pairwise nonisomorphic, Theorem 4.12 also yields that (b) implies (a). Next, if (c) is assumed, then
certainly VE has to be self-contragredient; what is left, in order to prove
the equivalence of the first set of conditions, is to show that (b) implies (c).
For this purpose we observe that, if VE is assumed to be self-contragredient,
then also all the other simple constituent of V E are self-contragredient, as
obviously we have ((VE )η )∗ ' ((VE )∗ )η for all η in Gal(E|F). Suppose now
that the form f¯ vanishes on a simple constituent of V E , say (VE )η ; since f¯
is nonsingular on V E , fixed an element v in (VE )η there must be an element
w lying in another simple constituent of V E , say (VE )ξ , such that f¯(v, w)
is not zero. But now f¯ provides an EG-isomorphism between (VE )η and
((VE )ξ )∗ (which is in turn isomorphic to (VE )ξ ), and this is a contradiction.
We conclude that f¯ does not vanish on any simple constituent of V E . This
also means that the orthogonal (with respect to f¯) of a simple constituent is
a direct complement for it. On the other hand, each simple constituent is a
homogeneous component and therefore it has a unique complement in V E ,
60
4. MODULES AND BILINEAR FORMS
namely the direct sum of the other homogeneous components. It follows
that the orthogonal of a simple constituent contains all the other simple
constituents, and the proof of the first part of the theorem is complete.
We move now to the second set of conditions; in order to prove that
0
(a ), (b 0 ) and (c 0 ) are equivalent, we only have to show that (c 0 ) follows
from the others. First of all, if VE is assumed to be not self-contragredient,
then all the simple constituents of V E are also not self-contragredient (this
follows from the fact that they all lie in a single Galois orbit). Now of
course f¯ has to vanish on all of them, and each has to be orthogonal (with
respect to f¯) to all the others except to the one that is contragredient to
it. We know that the contragredient of (VE )η is (VE )ητ , so let us bracket
in pairs the simple direct summands of V E , matching (VE )η with (VE )ητ
for all η in Gal(E|F). We get a direct decomposition of V E in which each
summand is the direct sum of two simple submodules on which f¯ vanishes
and which are contragredients of each other. Moreover, these two-component
summands are pairwise orthogonal to each other, and so the restriction of
the nonsingular f¯ to each of them must be nonsingular.
We are now in a position to draw a picture of the relationship between
simple modules and bilinear forms. Let G be a group, F a finite field, and
V a simple FG-module. Then we have three possibilities.
• V is not self-contragredient. In this case the discussion in 4.4 tells us
that it is impossible to define any G-invariant nonsingular bilinear F-form
on V . Of course the same happens on VE with E-linear forms, as now VE
is definitely not self-contragredient.
• V and VE are both self-contragredient. Assume first that the characteristic of F is odd. In this case, either V and VE can both carry G-invariant
nonsingualr symmetric forms but not symplectic or elusive ones, or V and
VE can both carry G-invariant nonsingular symplectic forms but not symmetric or elusive ones. For, by 4.5, V can be endowed with a G-invariant
nonsingular bilinear F-form, indeed with a symmetric or a symplectic one
(this last claim follows from Lemma 4.8). Since VE is self-contragredient,
by Theorem 4.13 we get that τV is the identity, and therefore all the
G-invariant forms on V have to be of the same type (see 4.7). The relevant
type is now the symmetric or the symplectic one. Let us assume that the
first option occurs, so we can choose a G-invariant nonsingular symmetric
F-form f on V . Now, the restriction to VE of the form f¯ (as defined in
introducing Theorem 4.13) is nonsingular and symmetric as well, and since
EQUIVALENCE OF FORMS
61
τVE is certainly the identity, also on VE we have only forms of the symmetric
type. The discussion is similar in the symplectic case. If the characteristic
of F is 2, then all the G-invariant nonsingular bilinear F-forms, both on V
and VE , are symmetric; indeed, they turn out to be all symplectic provided
V is not the trivial module (see Remark 4.15 together with [HB], VII, 8.13).
• V is self-contragredient, but VE is not. The situation is now that τ is
an involution (by Theorem 4.13) and, assuming odd characteristic for F, we
claim that all the three types of G-invariant nonsingular bilinear F-forms
appear on V , whereas no forms can be defined on VE . The second part of
the claim follows again by 4.4. As regards the first part, suppose we find a
symmetric form f on V (again, the argument is the same if we start from
a symplectic f ) and choose an element γ in E which is not fixed by τV ;
the form g defined by g(u, v) := f (u, vγ) is now certainly not symmetric
(see 4.7). If g is elusive, then we are done by Lemma 4.8. If g is symplectic,
then it is easy to see that the form h defined by h(u, v) := f (u, v)+g(u, v) is
nonsingular and elusive. This proves our claim. If F has characteristic 2, we
have that V carries G-invariant nonsingular F-forms of both the symmetric
and the elusive kind. As before, it will be clear from Remark 4.15 (and again
from [HB], VII, 8.13) that the symmetric forms are indeed all symplectic.
Equivalence of forms
We introduce now the concept of equivalent forms. In what follows we
refer to a particular type of forms (the symplectic one) in order to ease
the discussion, but keeping in mind that almost nothing changes in the
symmetric context.
4.14. Let G be a group, F a finite field, V a simple FG-module, and let f ,
g , h be G-invariant nonsingular symplectic F-forms on V ; if there exists
ε in AutFG (V ) such that g(u, v) = f (uε, vε) holds for all u and v in V ,
then we say that g is equivalent to f (and we write g ∼ f ) by means of ε.
This of course defines an equivalence relation on the set of G-invariant
nonsingular symplectic F-forms on V , and we have g ∼ f if and only
if the element γg of AutFG (V ), which corresponds to g in the bijection
yielded by f , is equal to εετ for some ε in AutFG (V ). If f is chosen as a
distinguished form, then we have g ∼ h if and only if γg γh−1 is an element
of the subgroup K of AutFG (V ) defined by K := {εετ : ε ∈ AutFG (V )}.
Of course the equivalence relation is not affected by the choice of f .
62
4. MODULES AND BILINEAR FORMS
Assume now the setting of 4.14; if p is the characteristic of F, then the
order of the field E = EndFG (V ) is pα for some integer α, and Aut(E) is
a cyclic group of order α whose elements are the pi -th powering maps on
E (i is an integer running from 0 to α − 1). Suppose that τ is not the
identity on E; in this case α is necessarily an even number, say 2β , and τ
is given by the pβ -th powering. Using that AutFG (V ) is a cyclic group of
order p2β − 1, we can now draw the following conclusion. if τ is not the
identity on E, then all the elements of E which are fixed by τ are equal to
εετ for some ε in AutFG (V ). Obviously, if τ is the identity on E, then the
elements of AutFG (V ) which (are fixed by τ and) are equal to εετ for some
ε in AutFG (V ) are precisely the squares of AutFG (V ).
As an immediate consequence of the previous discussion, we derive the
following
4.15. Remark. Let G be a group, F a finite field of characteristic 2, and
V a simple FG-module which carries a G-invariant nonsingular symplectic
F-form f . We know that any other form g on V , which is G-invariant
nonsingular and symmetric, is attached to an element γ of AutFG (V ) such
that γ is fixed by τ and g(u, v) = f (u, vγ) holds for all u, v in V . But
now, by the discussion above, any element in AutFG (V ) which is fixed by τ
is of the form εετ for some ε in AutFG (V ), no matter if τ is the identity
or not. What we get then is g(u, v) = f (uε, vε) for all u, v in V , so that
g is symplectic as well. We conclude that, in the present context, all the
G-invariant nonsingular symmetric F-forms on V are indeed symplectic.
And also the following
4.16. Proposition. Let G be a group, F a finite field, V a simple
FG-module which carries a G-invariant nonsingular symplectic F-form.
(a) If τ is not the identity on EndFG (V ), then there is a unique equivalence
class of G-invariant nonsingular symplectic F-forms on V ; this also
holds if τ is the identity on EndFG (V ), provided the characteristic of F
is 2.
(b) If τ is the identity on EndFG (V ), and F has odd characteristic, then
there are two equivalence classes of G-invariant nonsingular symplectic
F-forms on V ; if f and g are two such forms, and g is attached to the
element γ of AutFG (V ) via the bijection defined by f , then f and g
are equivalent if and only if γ is a square in AutFG (V ).
INDUCED FORMS
63
Induced forms
From now on, throughout this section, our setting will be the following:
G is a group, H is a subgroup of G whose index is n, F is a finite field, V
is a simple FG-module, and W is a submodule of V ↓H such that V ' W↑G .
4.17. Assume that f is an H -invariant nonsingular symplectic F-form defined on W . Denoting by {g1 , . . . , gn } a right transversal for H in G, we
g
g
define the map fgj : W gj × W gj → F by fgj (w1j , w2j ) := f (w1 , w2 ) for
all w1 , w2 in W (and for all j in {1, . . . , n}). Next, for u and v in V ,
consider the uniquely determined sequences {u1 , . . . , un } and {v1 , . . . , vn }
P
P
g
g
of elements in W such that u = nj=1 uj j and v = nj=1 vj j ; we define the
map f↑V : V × V → F by
f↑V (u, v) :=
n
X
g
g
fgj (uj j , vj j ).
j=1
V
It is easily checked that f↑ is a G-invariant nonsingular symplectic F-form
on V . Observe that, although the construction above involves the choice of
a transversal for H in G, the result of this construction is not affected at
all by such a choice, so that we can safely refer to f↑V as to the form on V
which is induced by f from W. Also, it is clear that the restriction of f↑V
to W is just the form f .
4.18. Let δ be an element of EndFH (W ); we define the map δ̄ on V by
P
v δ̄ := ni=1 (vi δ)gi , where {v1 , . . . , vn } is the uniquely determined sequence
P
g
in W such that v = ni=1 vi i . We see that δ̄ is in EndFG (V ); moreover, the
map from EndFH (W ) to EndFG (V ) which associates any δ in EndFH (W )
to δ̄ is a monomorphism of fields (observe that δ̄ is precisely the unique
element of EndFG (V ) whose restriction to W is δ : recall the universal
property 1.15). We shall not introduce a new notation for the copy of
EndFH (W ) which we recognized in EndFG (V ); it will be clear from the
context whether we refer to the abstract field or to its embedded copy.
Next, if τW is not the identity then τV is also not the identity: indeed, we
τ
see that δ τW = δ V for all δ in EndFH (W ).
4.19. Let f and g be H -invariant nonsingular symplectic F-forms on W ;
if γ is the element of EndFH (W ) such that g(w1 , w2 ) = f (w1 , w2 γ) for all
w1 , w2 in W , then we have g↑V (v1 , v2 ) = f↑V (v1 , v2 γ̄) for all v1 , v2 in V .
Moreover, if g is equivalent to f by means of an element ε of EndFH (W ),
then g↑V is equivalent to f↑V by means of ε̄.
64
4. MODULES AND BILINEAR FORMS
We are now in a position to change conveniently our point of view,
establishing a connection between the concept of form induction and the
concept of induced forms.
4.20. In our setting, assume that the FH -module W carries H -invariant
nonsingular symplectic F-forms (hence we are essentially in the hypothesis
of Conjecture 3.3). Using the construction of 4.17, we see that V carries
G-invariant nonsingular symplectic F-forms as well; chosen a form h , i
among them, we observe what follows (recall the definition of form induction (3.9)). If there exists an H -invariant nonsingular symplectic F-form
f on W such that h , i is equivalent to f ↑V , then V is form-induced
from H (with respect to the form h , i) by an FH -submodule isomorphic to W . More precisely, if ε is an element of EndFG (V ) such that
f ↑V (x, y) = hxε, yεi for all x, y in V , then V is form-induced by W ε
from H (this is an easy application of the definitions). Conversely, if h , i
is not equivalent to any form which is induced by W from H , then V
is not form-induced from H by any FH -submodule isomorphic to W ; indeed, for any submodule Z of V ↓H which is isomorphic to W , there exists
an element ε in EndFG (V ) such that Z = W ε (this follows again by the
universal property 1.15) and, if V is form-induced by W ε, then we can
define an H -invariant nonsingular symplectic F-form f on W , by setting
f (w1 , w2 ) := hw1 ε, w2 εi for all w1 , w2 in W . Now we see that the form f↑V
is equivalent to h , i by means of ε: for x and y in V , let {x1 , . . . , xn }
and {y1 , . . . , yn } be the uniquely determined sequences in W such that
P
P
g
g
x = nj=1 xj j and y = nj=1 yj j ; we get
n
n
n
n
X
X
X
X
gj
gj gj
hxε, yεi = (
xj )ε, (
yj )ε =
(xj ε) ,
(yj ε)gj
j=1
=
n
X
j=1
=
n
X
j=1
h(xj ε)gj , (yj ε)gj i =
j=1
n
X
j=1
hxj ε, yj εi
j=1
f (xj , yj ) = f↑V (x, y)
j=1
(in this chain of equalities we used that ε is in EndFG (V ), W ε is orthogonal,
with respect to the form h , i, to all its translates except itself, and h , i is
a G-invariant form).
The upcoming lemma (which requires an easy introductory proposition)
provides a very effective criterion to determine the existence of ‘bad’ forms
INDUCED FORMS
65
on the module V . Roughly speaking, a form on V is bad for us if it is not
equivalent to any form which is induced from W ; we shall see that such a
form exists if and only if there are two equivalence classes of forms on V
(hence the same happens on W ) and, in the process of induction, the two
classes of forms on W are merged together, so that the other class of forms
on V remains uncovered.
Since Proposition 4.16 ensures that, if the field F has characteristic 2,
we have in any case a unique equivalence class of forms, from now on we
can safely restrict our attention to the case in which the characteristic of F
is odd.
4.21. Proposition. Let K1 and K2 be finite fields of odd characteristic,
such that K1 is a subfield of K2 . Then the following conditions are equivalent:
(a) the degree of K2 over K1 (as a field extension) is an even number;
(b) there exists an element ξ in K2 \ K1 such that ξ 2 lies in K1 ;
(c) all the elements of K1 are squares in K2 .
Proof. Recall that, in the present setting, K2 is a Galois extension of
K1 , and the Galois group Gal(K2 |K1 ) is cyclic of order equal to |K2 : K1 |. If
|K2 : K1 | is assumed to be even, then Gal(K2 |K1 ) has a subgroup of index 2,
and the fundamental theorem of Galois Theory ensures that there exists a
subfield K of K2 , containing K1 , which has degree 2 as a field extension
of K1 . But now K can be realized as a simple quadratic extension of K1 ,
which means exactly that there exists an element ξ in K \ K1 such that
ξ 2 is in K1 . This proves that (a) implies (b); conversely, if we assume the
existence of an element ξ as in (b), then we get
|K2 : K1 | = |K2 : K1 (ξ)||K1 (ξ) : K1 | = 2|K2 : K1 (ξ)|.
It is also easily seen that (b) implies (c): indeed, ξ 2 is not a square in
K1 (otherwise ξ would lie in K1 ), whence the set of non-squares of K1 can
be realized as {ξ 2 ε : ε a square in K1 }, and each element of this set is now
clearly a square in K2 . It is totally straightforward that (c) implies (b), and
therefore our claim is proved.
4.22. Lemma. Let G be a group, H a subgroup of G, F a field of odd characteristic, V a simple FG-module, and W a submodule of V ↓H such that
V ' W ↑G . Assume that W carries H -invariant nonsingular symplectic
F-forms; then the following conditions are equivalent:
66
4. MODULES AND BILINEAR FORMS
(a) the automorphism τV is the identity on EndFG (V ), and the degree of
EndFG (V ) over EndFH (W ) (as a field extension) is an even number;
(b) there exists a G-invariant nonsingular symplectic F-form on V which
is not equivalent to any of the forms induced from W .
Moreover, for any G-invariant nonsingular symplectic F-form on V , there
exists a submodule of V ↓H , isomorphic to W , on which the relevant form
does not vanish.
Proof. Let us suppose that condition (a) holds. Since τV is the identity
on EndFG (V ), Proposition 4.16(b) ensures that the set of G-invariant nonsingular symplectic F-forms on V is partitioned into two equivalence classes,
say S1 and S2 . But also, by 4.18, τW is the identity on EndFH (W ), so that
there are two equivalence classes s1 and s2 of H -invariant nonsingular symplectic F-forms on W . Suppose that an element f of s1 yields an induced
form f↑V which lies in S1 : the discussion in 4.19 guarantees that any form
in s1 induces forms lying in S1 as well. Consider now an element g in s2 ,
and the element δ in AutFH (W ) such that g(w1 , w2 ) = f (w1 , w2 δ) holds
for all w1 , w2 in W ; we have g↑V (v1 , v2 ) = f↑V (v1 , v2 δ̄) for all v1 , v2 in
V and, although δ is not a square in AutFH (W ), δ̄ is indeed a square in
AutFG (V ) by the previous proposition. This yields that g↑V is equivalent
to f↑V , so that the process of induction ‘maps’ also the class s2 to the class
S1 . At this stage we see that, given a form in S2 , this can not be equivalent
to any of the forms induced from W .
Conversely, let us assume condition (b). Of course we must have two
equivalence classes of G-invariant nonsingular symplectic F-forms on V
(otherwise just consider a form f on W and induce it up; now all the forms
on V are equivalent to f ↑V ), whence τV is the identity. Moreover, let f
and g be inequivalent forms on W ; if δ is the element of AutFH (W ) such
that g(w1 , w2 ) = f (w1 , w2 δ) holds for all w1 , w2 in W , then we know that
also g ↑V (v1 , v2 ) = f ↑V (v1 , v2 δ̄) holds for all v1 , v2 in V , and δ̄ is not
a square in AutFH (W ). But our assumption forces δ̄ to become a square
in AutFG (V ), and now the previous proposition yields that the degree of
EndFG (V ) over EndFH (W ) as a field extension in an even number.
We move now to the last part of the statement. First of all, let f¯ and ḡ
be G-invariant nonsingular symplectic F-forms on V , and assume that X is
a submodule of V ↓H , isomorphic to W , on which f¯ does not vanish; assume
also that f¯ is equivalent to ḡ by means of the element ε in AutFG (V ). Then
it is easy to see that ḡ does not vanish on Xε. Moreover, any form on V
INDUCED FORMS
67
which is induced from W of course does not vanish on W ; therefore, we
only have to deal with the case in which there are forms on V not induced
from W , and it will be enough to prove the claim for one of them.
For this purpose, consider an H -invariant nonsingular symplectic F-form
f on W , and denote by h , i the form f↑V on V . We choose a generator, say
ζ , of AutFG (V ); it is clear that ζ is not a square in AutFG (V ), so we are sure
that the form [ , ] defined by [x, y] := hx, yζi for all x, y in V is not equivalent to any form which is induced from W ; moreover, EndFG (V ) can be
obtained as a simple extension of EndFH (W ) by adjoining ζ . Let us assume
that W γζ is orthogonal, with respect to h , i, to W γ for all γ in AutFG (V );
then, in particular, W ζ r−1 is orthogonal to W ζ r , and W (1+ζ r−1 ) is orthogonal to W (1 + ζ r−1 )ζ for all r in {1, . . . , n := |EndFG (V ) : EndFH (W )|}.
Therefore we have
0 = hw + wζ r−1 , zζ + zζ r i = 2hw, zζ r i
for all w, z in W (recall that in the present situation τV is the identity,
so that hwζ r−1 , zζi = hw, zζ r i holds). It follows that W is orthogonal to
W ζ r for all r in {1, . . . , n}. Now, there exists a sequence {δ0 , . . . , δn−1 } of
P
j
elements in EndFH (W ) such that ζ n = n−1
j=0 δj ζ holds. Hence, for all w, z
in W , we get
0 = hw, zζ n i = hw, zδ0 + zδ1 ζ + · · · + zδn−1 ζ n−1 i = hw, zδ0 i.
Since h , i does not vanish on W , δ0 has to be 0, which is a contradiction.
We conclude that there exists γ in AutFG (V ) such that W γ is not orthogonal to W γζ ; now it is clear that [ , ] does not vanish on W γ , and our
claim is proved.
It would be very desirable, for our purposes, to argue that the situation
outlined in the previous lemma can not arise at all, if we assume that H has
odd index in G. Unfortunately, as Example 6.2 will show, such a situation
may very well occur even if |G : H| is an odd prime.
CHAPTER 5
Positive results
In Chapter 3 and Chapter 4, following a chain of reductions, we departed
quite a lot from our original problem on tensor induction for complex representations (Conjecture 3.2). Now it is time to approach the last ‘link’ of
this chain, which is Conjecture 4.1; drawing some positive conclusions about
it will immediately enable us to go back, and show that all the earlier conjectures (with some extra hypotheses) are indeed theorems. Namely, if we
focus on induction from normal subgroups, the weak versions of Conjecture
3.2 and Conjecture 3.3 get a full answer (which is positive for subgroups of
odd index, and negative in general for subgroups of even index; see Theorem
5.23 and Theorem 5.24, together with Example 3.18); as regards subgroups
which are not necessarily normal, we don’t have such a conclusive result, but
Conjectures 3.2 and 3.3 are proved (of course, in the weak version) for subgroups of (odd) prime index (see Theorem 5.20 and 5.21). It appears very
much conceivable, at this stage, that extending 5.20 and 5.21 to subgroups
whose index is a power of an odd prime would provide a really conclusive
answer. This is left for the future.
Induction from maximal subgroups
The main results of this section are Theorem 5.14 and Theorem 5.18;
roughly speaking, both of them concentrate on a situation in which a simple
FG-module V is induced from a maximal subgroup H of G (of course the
setting will be in both cases much more specific), and their purpose is to
achieve some control on the structure of V ↓H . With such a control, as we
shall see in the following section, it is easy to draw finally some positive
conclusions towards our conjectures.
Some preparation is needed before we get to 5.14 and 5.18; the critical
steps in such a preparation will be essentially Lemma 5.3 (in turn preceded
by two obvious remarks, whose aim is to provide some more comfort in the
reading of 5.3), and Lemma 5.11.
69
70
5. POSITIVE RESULTS
5.1. Remark. Let G be a group, F a field, U an FG-module, and V , Z
submodules of U such that U = V ⊕ Z . The projection of U on the direct
summand (say) V is the map pV defined as follows: for any u in U , let v
and z be the uniquely determined elements, of V and Z respectively, such
that u = v + z ; then pV maps u to v .
As it is well known, pV is a surjective homomorphism of F-vector spaces
from U to V , but it is obviously a homomorphism of FG-modules as well.
In particular, assume that W is a submodule of U such that HomFG (W, V )
is the zero space; then W is contained in Z , since otherwise the restriction
of pV to W would provide a nonzero module homomorphism from W to
V (this argument can be used to prove that, in a semisimple module, each
homogeneous component contains exactly the submodules belonging to a
particular isomorphism type, and also that it has a unique complement).
5.2. Remark. Let G be a group, F a field, U an FG-module, and W
a submodule of U which is a direct summand. If V is a submodule of
U such that W lies in V , then W is also a direct summand of V ; more
precisely, if Y is a submodule of U such that U = W ⊕ Y , then we have
V = W ⊕ (V ∩ Y ).
Consider now our usual setting: G is a group, H a subgroup of G, F a
finite field, V a simple FG-module, and W a submodule of V ↓H which induces V from H ; the following lemma shows that the degree of EndFG (V )
as a field extension of EndFH (W ) has a precise meaning in terms of the
internal module structure of V ↓H . Although it can be derived from Theorem 4.12b) in [HB], VII, we prefer to give a complete proof of Lemma 5.3,
emphasizing some aspects of the situation which are relevant for us. Since
the universal property of induced modules mentioned in 1.15 is a critical
ingredient in the proof, it is worth recalling it briefly: in the given setting,
for any element α of HomFH (W, V ↓H ) there exists a unique element ᾱ in
EndFG (V ) such that ᾱ↓W = α.
5.3. Lemma. Let G be a group, H a subgroup of G, F a finite field,
V a simple FG-module, and W a submodule of V ↓H such that V ' W↑G .
Denoting by n the degree of EndFG (V ) as a field extension of EndFH (W ),
there exist submodules U and Y of V ↓H with the following properties:
(a) V ↓H = U ⊕ Y ;
(b) HomFH (W, Y ) = 0;
(c) U is semisimple, and all its simple constituents are isomorphic to W ;
INDUCTION FROM MAXIMAL SUBGROUPS
71
(d) the composition length of U is n.
Any submodule U 0 of V ↓H which satisfies (c) is contained in U ; equality occurs if and only if there exists a submodule Y 0 of V ↓H such that
V ↓H = U 0 ⊕ Y 0 and HomFH (W, Y 0 ) = 0. Moreover, Y is the unique direct
complement for U in V ↓H .
Proof. As the first step we shall show that, for any integer k in
{1, . . . , n}, there exist a submodule Yk of V ↓H and a sequence (indeed
a set) {α1 , . . . αk } of nonzero elements in HomFH (W, V ↓H ), such that
L
V ↓H = ( ki=1 W αi ) ⊕ Yk . Since V is induced by W , which is a submodule of V ↓H , we have that W is a direct summand of V ↓H (see 1.18(b)),
hence the claim is verified for k = 1. Assume now k greater than 1. By
inductive hypothesis we find a submodule Yk−1 of V ↓H , and a sequence
{α1 , . . . , αk−1 } in HomFH (W, V ↓H ), such that
V ↓H =
k−1
M
W αi ⊕ Yk−1 .
i=1
Suppose for the moment that HomFH (W, Yk−1 ) = 0, and let β be an arbitrary element of HomFH (W, V ↓H ). By Remark 5.1 we get
Wβ ≤
k−1
M
W αi ;
i=1
therefore, for any given w in W , there exist uniquely determined elements
w1 , . . . , wk−1 in W such that wβ = w1 α1 + · · · + wk−1 αk−1 holds. It is
easily checked that the maps εj , defined on each w in W by wεj := wj ,
are elements of EndFH (W ); it follows that β = ε1 α1 + · · · + εk−1 αk−1 , and
of course β = ε1 α1 + · · · + εk−1 αk−1 (we are using the notation coming
from the universal property 1.15). Since this can be done with any β in
HomFH (W, V ↓H ), the elements α1 , α2 , . . . , αk−1 span EndFG (V ) as a
vector space over EndFH (W ), which is not the case. We conclude that
there exists a nonzero element αk in HomFH (W, Yk−1 ) and, since W αk is
a direct summand of V ↓H which is contained in Yk−1 , by 5.2 it is also a
direct summand of Yk−1 . This concludes the first step of the proof, giving
Ln
us the modules U :=
i=1 W αi , and Y := Yn ; certainly they provide a
direct decomposition for V ↓H , as asserted in (a).
It is clear that the module U , as constructed, satisfies conditions (c)
and (d). Assume that (b) is not satisfied by Y ; if this were the case, we
could choose a nonzero element αn+1 in HomFH (W, Y ), and then add the
72
5. POSITIVE RESULTS
direct summand W αn+1 to U . But this leads to a contradiction, since
now α1 , . . . , αn+1 turn out to be linearly independent over EndFH (W ).
Indeed, assume that ε1 , . . . , εn+1 are elements in EndFH (W ) such that
ε1 α1 + · · · + εn+1 αn+1 = 0 holds; in particular we have
wε1 α1 + · · · + wεn+1 αn+1 = 0
for all w in W and, as wεj αj is in W αj , we get wεj αj = 0 for all w in
W . It follows that εj αj is zero, hence εj and εj are zero. We conclude that
condition (b) holds for Y , and now we can also observe that {α1 , . . . , αn }
is a basis for EndFG (V ) over EndFH (W ).
We move now to the uniqueness part of the statement. Assume that U 0
is a submodule of V ↓H which satisfies (c); each simple FH -submodule of
U 0 must project trivially on Y (again by 5.1), so that U 0 is contained in
U . If there exists a submodule Y 0 of V ↓H such that V ↓H = U 0 ⊕ Y 0 and
HomFH (W, Y 0 ) = 0, of course we get U ≤ U 0 by the same argument. Let us
show that, in such a situation, Y 0 coincides with Y as well. Suppose that
there exists a direct summand of U , say W αr , such that the projection p
of Y 0 on W αr is not zero. If we compose the projection of V ↓H on Y 0
with p, we obtain a map π which is an epimorphism of V ↓H to W αr ; since
the codimension of ker π in V ↓H is given by (dim V )/|G : H|, recalling the
discussion after 1.18 we see that ker π is a direct summand of V ↓H . But
now we get
V ↓H = ker π ⊕ Z
where Z is of course a submodule of V ↓H isomorphic to W , and this leads
to a contradiction, since U is contained in ker π and at the same time it
contains Z . We conclude that Y 0 projects trivially on each direct summand
of U , and therefore it is contained in Y ; but clearly equality holds, since
dim Y 0 is the same as dim Y .
Before going through Lemma 5.7, we need to introduce some terminology.
5.4. Definition. Let G be a group, F a field, and V an FG-module.
Consider the F-subspace of V given by the sum of all the simple submodules
of V ; such a subspace is indeed a submodule, called the socle of V and
denoted by soc(V ). It is clear that soc(V ) is a semisimple module which
contains all the semisimple submodules of V .
INDUCTION FROM MAXIMAL SUBGROUPS
73
5.5. Remark. Observe that the module U which appears in the statement
of Lemma 5.3 is precisely the homogeneous component of soc(V ↓H ) containing W , and part of the conclusions of 5.3 may be paraphrased saying
that the degree of EndFG (V ) as a field extension of EndFH (W ) is the composition length of this homogeneous component, which also turns out to be
a direct summand of the whole V ↓H (and not only of soc(V ↓H )).
5.6. Definition. Let G be a group, F a field, and X an FG-module; then
by ker X we denote the subgroup of G constituted by the elements which
act trivially on X . Observe that, if D is a representation associated to X ,
then ker X is precisely ker D .
5.7. Lemma. Let H be a group, L a normal subgroup of H , F a finite
field, and X a 1-dimensional FH -module whose kernel contains L. Let W
be a simple FH -module. Then W ⊗ X and W have the same (nonzero)
multiplicity as composition factors in the socle of W↓L↑H .
Proof. First of all, by Nakayama reciprocity (1.14) we have
HomFH (W ⊗ X, W↓L↑H ) ' HomFL ((W ⊗ X)↓L , W↓L ) '
' HomFL (W↓L , W↓L ) ' HomFH (W, W↓L↑H )
(here we are dealing with vector space isomorphisms). Moreover, since X is
1-dimensional, it is not hard to check that EndFH (W ⊗ X) is isomorphic,
as a vector space, to EndFH (W ). Denoting by m(Z) the multiplicity of
a simple FH -module Z as a composition factor in the socle of W ↓L↑H ,
Theorem 4.12b) in [HB], VII, gives now
m(W ⊗ X) =
dimF HomFH (W ⊗ X, W↓L↑H )
=
dimF EndFH (W ⊗ X)
dimF HomFH (W, W↓L↑H )
=
= m(W ),
dimF EndFH (W )
as desired.
Propositions 5.8-5.10 will prepare the setting for the critical Lemma 5.11.
In what follows, we use some notation and concepts introduced in the first
section of Chapter 1 (more precisely, in 1.4-1.8).
5.8. Proposition. Let G be a group, F a field, U and V FG-modules,
and K a field extension of F; then we have
HomKG (U K , V K ) ' HomFG (U, V ) ⊗ K,
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5. POSITIVE RESULTS
where the symbol ‘'’ denotes an isomorphism of K-vector spaces (the relevant structure of K-vector space on HomFG (U, V ) ⊗ K is outlined -for finite
degree extensions- in 1.5). In particular, dimK HomKG (U K , V K ) is the same
as dimF HomFG (U, V ).
For the proof of 5.8, see [Hu], V, 11.9.
5.9. Proposition. Let G be a group, H a subgroup of G, F a field, and
K a field extension of F; also, let U ,V ,Z be FG-modules, and W an
FH -module. Then the following properties hold:
(a)
(b)
(c)
(d)
U ⊗ (V ⊕ Z) ' (U ⊗ V ) ⊕ (U ⊗ Z);
(U ⊕ V )K ' U K ⊕ V K ;
(V ↓H )K ' (V K↓H );
(W↑G )K ' (W K )↑G .
Proof. This can be easily checked by comparing the actions of the relevant group on the two sides (recall that, if {x1 , . . . , xn } is an F-basis
for a module X , then {x1 ⊗ 1, . . . , xn ⊗ 1} provides a K-basis for X K
(see 1.5)).
5.10. Proposition. Let G be a group, F a splitting field for G, and V a
semisimple FG-module. Assume that V has k homogeneous components,
and that the composition length of the i-th homogeneous component is ni
for all i in {1, . . . , k}. Then EndFG (V ) is isomorphic (as an F-algebra) to
Lk
i=1 Mat(ni , F), where Mat(ni , F) denotes the algebra of ni × ni matrices
with entries in F.
(See Lemma 3 in [Al], Chapter 1).
5.11. Lemma. Let H be a group, L a normal subgroup of H , F a finite
field of odd characteristic, and W an absolutely simple FH -module. Assume
that there exists an FH -module X such that its kernel M contains L,
|H : M | = 2, and W ⊗ X is isomorphic to W . Then the multiplicity of W
as a composition factor in the socle of W↓L↑H is a positive even number.
Proof. As the first step, we shall prove the following claim: suppose that
it is possible to find a field extension K of F, having finite degree over F,
such that the lemma holds with K in place of F; then the lemma holds for
F as well.
Let K be such an extension, and consider the module W K ; this is simple, as W is absolutely simple (see 1.6), but W K is also absolutely simple
INDUCTION FROM MAXIMAL SUBGROUPS
75
because, by Proposition 5.8, we have
dimK EndKH (W K ) = dimF EndFH (W ) = 1.
Consider now the regular module F[H/M ]: since the characteristic of F
is odd, by Maschke’s Theorem (1.2) F[H/M ] is semisimple; moreover, it
contains as a direct summand each simple F[H/M ]-module (see [Is], 9.5(a)),
whence it is the direct sum of X and the 1-dimensional trivial FH -module
(both regarded in a natural way as F[H/M ]-modules). By 1.23, 5.9(a), and
our assumption W ' W ⊗ X , we get
W↓M↑H ' W ⊗ F[H/M ] ' W ⊕ W,
and therefore, by 5.9(b,c,d)
(W K )↓M↑H ' (W↓M↑H )K ' (W ⊕ W )K ' W K ⊕ W K .
On the other hand, as before, we have (W K )↓M↑H ' W K ⊗ K[H/M ], and of
course K[H/M ] is the direct sum of the 1-dimensional trivial KH -module
and another 1-dimensional KH -module, say X 0 (which is indeed X K ),
whose kernel is M ; we conclude that W K ⊗ X 0 is forced to be isomorphic to W K . Now, as we are assuming that the lemma holds for K, we
have that the multiplicity of W K in the socle of (W K )↓L↑H is a positive
even number. But, as we observed in proving Lemma 5.7, this multiplicity
is given by dimK HomKH (W K , (W K )↓L↑H ), and we have
dimK HomKH (W K , (W K )↓L↑H ) = dimK HomKH (W K , (W↓L↑H )K )
= dimF HomFH (W, W↓L↑H ).
The claim is now proved, because the last member in this chain of equalities
is indeed the multiplicity of W in the socle of W↓L↑H .
Since we know that there exists a finite degree field extension of F which
is a splitting field for G and for all its subgroups (see 1.8), by the previous
step we can certainly concentrate on the case in which F itself is such a
splitting field; this will be henceforth our assumption.
Observe that, as the multiplicity of W in W↓M↑H is 2, we get
dimF EndFM (W↓M ) = dimF HomFH (W, W↓M↑H ) = 2.
Now, W↓M is certainly not simple (otherwise, since F is a splitting field for
M , the dimension of EndFM (W↓M ) would be 1), and it is not homogeneous
with composition length 2 as well (in this case, by Proposition 5.10, we would
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5. POSITIVE RESULTS
get EndFM (W↓M ) ' Mat(2, F), and its dimension over F would be 4). The
only other possibility is
W↓M = Y ⊕ Y h ,
where Y is a simple FM -module, h is in H \M , and Y h is not isomorphic to
Y . This tells us that the composition length of W↓L is an even number, since
it is twice the composition length of Y ↓L (by the way, in order to achieve
this conclusion, it was relevant just to exclude that W↓M were simple). On
the other hand, assume that W ↓L has k homogeneous components, each
having composition length m: the composition length of W↓L is now given
by km, and we know that this is an even number. But EndFL (W ↓L ) is
isomorphic to the direct sum of k copies of Mat(m, F), hence its dimension
over F is km2 , an even number as well; since this is also the dimension over
F of HomFH (W, W↓L↑H ), which is in turn the multiplicity of W in the socle
of W↓L↑H , the proof is complete.
We move now to the following situation: G is a solvable group, H a
subgroup of G having odd prime index, F a finite field of odd characteristic, and V a simple FG-module induced from H . Theorem 5.14 and
Theorem 5.18 (which indeed requires a weaker hypothesys for H ) provide
some information, concerning the structure of V ↓H , which will be extremely
useful for our purposes.
5.12. Remark. Suppose that G is a solvable group, H is a non-normal
subgroup of G having prime index, and L is the kernel of the permutation
action of G (by right multiplication) on the set of right cosets modulo H .
As permutation group on this set, G/L is then a Frobenius group of prime
degree, with H/L as Frobenius complement (see [Hu2], §16); denote the
Frobenius kernel of G/L by K/L. Note that K/L is a normal Sylow subgroup of prime order and H/L is a complementary Hall subgroup for it in
G/L. Any two distinct conjugates of H/L have trivial intersection; moreover, each nontrivial subgroup of G/L either contains K/L or is contained
in a unique conjugate of H/L. We shall also use that any transversal of L in
K is a right transversal of H in G, that the permutation action of H on the
set of nontrivial cosets of G modulo H matches the conjugation action on
the nontrivial elements of K/L, and that the H -orbits of nontrivial cosets
all have the same length, namely |H/L| (= |G/K|).
5.13. Lemma. Let G be a solvable group, H a subgroup of G having odd
prime index, F a finite field of odd characteristic, V a simple FG-module,
INDUCTION FROM MAXIMAL SUBGROUPS
77
and W an absolutely simple submodule of V ↓H such that V ' W ↑G .
If W is not induced from the normal core L of H in G, then we have
L
V ↓H ' ( si=1 W ) ⊕ Y , where s is an odd number and Y is a submodule of
V ↓H such that HomFH (W, Y ) = 0.
Proof. First of all, observe that H is definitely not a normal subgroup
of G (otherwise W would be induced from L = H ). Let X be a simple
constituent of W↓L , and IG (X) its inertia subgroup in G (which certainly
contains L). By 5.12, there are three cases to distinguish. First, if there
exists an element g in G \ H such that L ≤ IG (X) ≤ H g , then we get
IH (X) = L and therefore W is induced by X from L, so this case can not
arise. Second, we shall show that, if L < IG (X) ≤ H , then every simple
submodule of V ↓H other than W itself has strictly larger dimension than
W . In view of 5.3, this will prove our claim for this case (with s = 1). Start
by noting that now I := IH (X) = IG (X) and we have IH (X g ) = H ∩I g = L
for all g in G \ H . Consider a simple submodule T of V ↓H , and suppose
first that T ↓L has a simple constituent, say Z , isomorphic toX h for some
h in H . Then X h and Z are in the same homogeneous component of V ↓L
and this homogeneous component, call it U , is simple as F[I h ]-module.
The F[I h ]-modules W ↓I h and T ↓I h both have nonzero intersection with
the simple F[I h ]-module U , so they both have to contain U . Thus the
simple FH -modules W and T have nonzero intersection, and this implies
W = T . We are only interested in T if this does not hold, and now we
know that then no simple constituent of T↓L can be isomorphic to an X h .
Of course, any simple constituent Z of T↓L (indeed, any simple constituent
of V ↓L ) is isomorphic to X g for some g in G; the conclusion from our
argument so far is that Z ' X g for some g in G \ H . Thus IH (Z) is
L, so that T is induced by Z from L. From this, we can see that the
dimension of T is greater than the dimension of W : otherwise we would
get dim W = dim T = |H : L| dim Z = |H : L| dim X , and therefore W
would be induced by X from L.
We are left with the case in which IG (X) is not contained in any conjugate of H ; the structure of G, as it was outlined in Remark 5.12, forces
now IG (X) to contain the normal subgroup K (as defined in 5.12), so
that K is of course contained in all the conjugates of IG (X) and therefore it stabilizes all the simple constituents of W ↓L . In particular, since a
transversal for H in G can be built up using only elements of K , we get
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5. POSITIVE RESULTS
W y↓L↑H ' (W↓L )y ↑H ' W↓L↑H for all y in such a transversal. Now, considering the structure of G and taking in account the last comment, Mackey’s
Lemma yields
n
M
V ↓H ' W ⊕
W↓L↑H
i=1
where n is given by (|G : H| − 1)/|H : L| (this follows again from the
discussion in 5.12) . Let α be a nonzero element in HomFH (W, W↓L↑H );
W α is an FH -submodule of W ↓L↑H , in particular of V ↓H , so that it is a
direct summand in W ↓L↑H . Denoting by S a direct complement for W α
in W ↓L↑H , again we consider HomFH (W, S) and we iterate the process,
L
eventually getting W↓L↑H ' ( ti=1 W ) ⊕ R, where R is an FH -submodule
of W↓L↑H such that HomFH (W, R) = 0 (observe that t is now precisely the
multiplicity of W in the socle of W↓L↑H ). Therefore we get
V ↓H '
nt+1
M
W ⊕Y
i=1
Ln
where Y is defined as
i=1 R, and of course we have HomFH (W, Y ) = 0.
Our aim is now to show that nt is an even number. Certainly it is such if
n is even. If n is odd then |H : L| has to be even, and we shall see that in
this case t turns out to be even.
Assume then n odd, and consider the representation of H/L which
maps a generator to −1 in F; we claim that, if X denotes an FH -module
associated to this representation, then W ⊗ X is isomorphic to W . Indeed,
by Lemma 5.7, W ⊗ X and W have the same multiplicities as composition
factors in the socle of W ↓L↑H . If they are assumed to be nonisomorphic,
this implies
nt
nt+1
M
M V ↓H '
W ⊕Y '
(W ⊗ X) ⊕ Y 0 ,
i=1
i=1
0
where HomFH (W, Y ) = HomFH (W ⊗ X, Y ) = 0. But now, recalling that
EndFH (W ) and EndFH (W ⊗X) are isomorphic vector spaces, Lemma 5.3(a)
gives
nt + 1 = |EndFG (V ) : EndFH (W )| = |EndFG (V ) : EndFH (W ⊗ X)| = nt,
clearly a contradiction. We are now in a position to apply Lemma 5.11 (as
of course the kernel of X has index 2 in G), and the proof is complete.
The next step shall be to remove the hypothesis of absolute irreducibility
for W ; this is the purpose of the following theorem.
INDUCTION FROM MAXIMAL SUBGROUPS
79
5.14. Theorem. Let G be a solvable group, H a subgroup of G having odd
prime index, F a finite field of odd characteristic, V a simple FG-module,
and W a submodule of V ↓H such that V ' W ↑G . If W is not induced
L
from the normal core L of H in G, then we have V ↓H ' ( si=1 W ) ⊕ Y ,
where s is an odd number and Y is a submodule of V ↓H such that
HomFH (W, T ) = 0.
Proof. Let us denote by E the field EndFG (V ), and by K the field
EndFH (W ). As we mentioned in the previous chapter, a structure of
EG-module can be defined on V by considering the natural action of E;
similarly, W acquires a structure of KH -module if we let K act naturally
on it; recall that the modules which arise in this way are denoted by VE and
WK respectively.
Also, K is embedded in E by means of a field monomorphism which
arises from the universal property 1.15 (see 4.18). Therefore, the underlying
vector space of V can be endowed in a natural fashion with a structure of
KG-module; we shall denote this module by VK . Since W is a subset of
VK (let i be the inclusion map of W in VK ), and the embedding of K in E
is such that (wi)(k̄) = (wk)i for all w in W and k in K (here k̄ denotes
the image of k under the relevant embedding), we conclude that WK is a
submodule of (VK )↓H . It is clear that VK is a simple KG-module, and WK
an absolutely simple KH -module; moreover, we have
dimK (VK ) =
|G : H|dimF W
dimF V
=
= |G : H|dimK (WK ),
|K : F|
|K : F|
whence we get VK ' (WK ) ↑G
H . We claim now that WK is not induced
from L. Assume WK ' X ↑H
L , where X is a submodule of (WK )↓L . Certainly W contains XF as an FH -submodule and, since
dimF (XF ) = dimK ((XF )K ) = |K : F|dimK X
holds, we have
dimF W = |K : F|dimK (WK ) = |K : F||H : L|dimK X = |H : L|dimF (XF ),
so that W is induced by XF from L, a contradiction.
We are finally in a position to apply Lemma 5.13, getting
s
M
(VK )↓H '
WK ⊕ Y,
i=1
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5. POSITIVE RESULTS
where s is an odd number and Y is a submodule of (VK ) ↓H such that
HomKH ((WK ), Y ) = 0. Now Lemma 5.3 yields
|EndKG (VK ) : EndKH (WK )| = s
and, since EndKG (VK ) is easily seen to be E, another application of
Lemma 5.3 leads to the desired conclusion.
At this stage, keeping in mind Lemma 5.3, it is clear that Conjecture 4.1
holds if we make two extra assumptions: H has prime index in G, and W
is not induced from the normal core of H ; our aim is now to get rid of the
latter.
If, in the setting of Theorem 5.14, we drop the assumption of W not
being induced from L, then we do loose control on the multiplicity of W
in the socle of V ↓H (see Chapter 6); at any rate, as it is shown in the next
lemma, we still have a good understanding (in some sense even a better one)
of the structure of V ↓H .
5.15. Lemma. Let G be a solvable group, H a maximal subgroup of G
having odd index, and F an arbitrary field; let V be a simple FG-module
and W a submodule of V ↓H such that V ' W ↑G . If W is induced from
the normal core L of H in G, then V ↓H is semisimple, and all its simple
submodules have dimension equal to dimW . In particular, the composition
length of V ↓H is an odd number.
Proof. By the solvability of G and the maximality of H , it is possible
to find a normal subgroup K of G such that HK = G and H ∩ K = L.
Consider now a submodule Y of W ↓L such that W is induced by Y from
L; we have
V ' (Y ↑H )↑G ' (Y ↑K )↑G .
If we view V ↓H as [(Y ↑K )↑G ]↓H , then Mackey’s Lemma yields
V ↓H ' ((Y ↑K )↓L )↑H .
Chosen a right transversal T for L in K (of course the cardinality of T is
|G : H|), we get now
M g H
[(Y )↑ ].
V ↓H '
g∈T
g
H
Since V is simple and the (Y )↑ all have minimal dimension, V is induced
from H by each of those; in particular they are all simple, so that V ↓H
is semisimple with all the simple submodules having the same (minimal)
dimension.
INDUCTION FROM MAXIMAL SUBGROUPS
81
With Lemma 5.15 we are in a position to prove Conjecture 4.1 also in the
case that W is induced from the normal core of H in G. In this situation
we do not even need to assume that the index of H in G is a prime; it
will be sufficient to require that H is a maximal subgroup. Observe that
also the hypothesis that W carries an H -invariant nonsingular symplectic
F-form is not relevant (it is enough that V is a symplectic module), and
characteristic 2 for the field F is allowed.
Before going through Theorem 5.18, we need to prove a general lemma
on symplectic modules; to make this diversion more complete, we shall pair
Lemma 5.16 with an example.
5.16. Lemma. Let H be a group, F a finite field, V a semisimple homogeneous FH -module, and f an H -invariant nonsingular symplectic F-form
on V . If the composition length of V is an odd number, then there exists a
simple submodule of V on which f does not vanish.
Proof. We proceed by induction on the composition length of V , which
we denote by lH (V ) and write as 2k + 1 (k is a nonnegative integer).
If k is 0, then the claim is certainly true. Assume then k > 0: if the
restriction of f to all the simple submodules of V is not zero, then of course
we are done (by the way, we shall see in Theorems A.7 and A.8 that this
can not happen); hence we can choose a submodule U of V on which f
vanishes, and which is maximal subject to satisfy this property. Now, we
have U ≤ U ⊥ , (here orthogonality is of course meant with respect to f ),
and since lH (V ) = lH (U ) + lH (U ⊥ ) holds (see 3.5), one among lH (U ) and
lH (U ⊥ ) is even and the other is odd. This ensures that U is properly
contained in U ⊥ and, if we denote by R a complement for U in U ⊥ , we
see that lH (R) = lH (U ⊥ ) − lH (U ) is also an odd number which is strictly
smaller than lH (V ). If f is singular on R then, denoting R ∩ R⊥ by D ,
we have that U ⊕ D is an FH -submodule of V on which f vanishes. Since
U is strictly contained in U ⊕ D , and the latter is strictly contained in V ,
this contradicts the hypothesis of maximality on U . We conclude that f↓R
is nonsingular, and now the claim follows by induction.
It is natural to ask whether, in the previous lemma, it is necessary to
assume odd composition length for V ; indeed, as the following example
shows, Lemma 5.16 fails as soon as that assumption is dropped.
5.17. Example. Let H be a group, F a field, and V an FG-module.
Consider an F-basis B = {v1 , . . . , vn } for V , and let B ∗ = {ν1 , . . . , νn } be
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5. POSITIVE RESULTS
the corresponding dual basis on V ∗ ; if D is the representation associated
to the module V with respect to the given basis B , then we know that the
action on V ∗ of any element h in H is expressed, with respect to the basis
B ∗ , by the matrix (D(h)t )−1 (the transpose inverse of D(h)).
Consider now the FH -module V ⊕V ∗ and, using {v1 , . . . , vn , ν1 , . . . , νn }
as a basis for it, define a form f on V ⊕ V ∗ extending by bilinearity the
following relations:
f (vi , vj ) = f (νi , νj ) = 0;
f (vi , νj ) = δij = −f (νj , vi )
for all i, j in {1, . . . , n} (here δij is meant to be 1 if i = j , and 0 if i 6= j ).
Certainly f is a nonsingular symplectic form on V ⊕ V ∗ ; moreover, we see
that it is also H -invariant. Indeed, let Φ be the matrix associated to f
with respect to the basis {v1 , . . . , vn , ν1 , . . . , νn }; it is easily checked that,
for all A in GL(n, F), we get
(At ⊕ A−1 )Φ(A ⊕ (At )−1 ) = Φ
(given two matrices M and N , we denote by M ⊕ N the block-diagonal
matrix, of dimension equal to dimM + dimN , which has M in the top-left
corner and N in the bottom-right corner). Since now any element h of H
acts on V ⊕ V ∗ as the matrix D(h) ⊕ (D(h)t )−1 , our claim is proved.
It is clear that, in order to disprove Lemma 5.16 in the even composition length case, it is sufficient to start from an FH -module which is
self-contragredient and of odd dimension; indeed, this yields that V ⊕ V ∗
is a homogeneous FH -module (with composition length 2) which carries
an H -invariant nonsingular symplectic F-form f , but of course f vanishes
on all the simple submodules (recall that no nonsingular symplectic form is
allowed on a space of odd dimension). Certainly any trivial module of odd
dimension will do; as a less ‘trivial’ example, consider A4 in the role of H ,
and let F be any field of odd characteristic. Denoting by K the (normal) Sylow 2-subgroup of H , and choosing a nontrivial 1-dimensional FK -module
X , we have that V := X↑H is a 3-dimensional simple FH -module. Moreover, as it is easily seen, V carries an H -invariant nonsingular symmetric
form, so that V is self-contragredient.
5.18. Theorem. Let G be a solvable group, H a maximal subgroup of G
having odd index, F a finite field, V a simple FG-module which carries a
G-invariant nonsingular symplectic F-form f , and W a submodule of V ↓H
such that V ' W ↑G . Assume that W is induced from the normal core L
INDUCTION FROM MAXIMAL SUBGROUPS
83
of H in G. Then there exists a submodule Z of V ↓H such that f does not
vanish on Z , V ' Z↑G , and | EndFG (V ) : EndFH (Z)| is an odd number.
Proof. Since, by Lemma 5.15, V ↓H is semisimple and its composition
length is an odd number, there exists an odd number d such that an odd
number of homogeneous components in V ↓H have composition length equal
to d. Since V carries a G-invariant nonsingular symplectic F-form, we have
that V is self-contragredient, and of course the same holds for V ↓H . The
‘dualization’ induces a permutation of order 2 on the set of homogeneous
components of V ↓H and, since it preserves the dimensions, it permutes indeed the set of homogeneous components with composition length d. But
now, since that set contains an odd number of elements, the relevant permutation has to fix some element in it; we conclude that there exists a
homogeneous component X in V ↓H such that X is self-contragredient,
and the composition length of X is odd. Of course any simple submodule
of X is also self-contragredient and, as we shall see, this implies that the
form f is nonsingular on X . Indeed, let R be the unique complement for
X in V ↓H ; given an element y in X , the restriction to R of the map yψf
is an element of R∗ (recall that yψf maps an element v in V to f (v, y);
see 4.3), and the map y 7→ (yψf )↓R provides a morphism of FH -modules
(call it β ) from X to R∗ . If f is assumed singular on X , then there exists
a nonzero element x in X ∩ X ⊥ (orthogonality is meant with respect to
f ), and clearly we have xβ 6= 0, otherwise x would lie in V ∩ V ⊥ which
is zero; we conclude that, decomposing X and R∗ into the direct sum of
simple submodules, say
X = U1 ⊕ · · · ⊕ Ud and R∗ = S1 ⊕ · · · ⊕ Sl ,
there exist i in {1, . . . , d} and j in {1, . . . , l} such that β↓Ui pSj is not the
zero map (here pSj denotes the projection of R∗ on Sj ). But now Ui is
isomorphic to Sj , and this is a contradiction because R∗ does not contain
any simple submodule isomorphic to Uj∗ .
Now, by Lemma 5.16, there exists a simple submodule Z of X on which
f does not vanish. Moreover, Z induces V from H (because of its dimension) and, by Lemma 5.3, we also have | EndFG (V ) : EndFH (Z)| = d, an
odd number.
84
5. POSITIVE RESULTS
Some positive answers to the conjectures
We are finally in a position to prove that Conjecture 4.1 is true, with
the extra assumption that the index of H in G is a prime number.
5.19. Theorem. Let G be a solvable group, H a subgroup of G having odd
prime index, F a finite field of odd characteristic, V a simple FG-module,
and W a submodule of V ↓H such that V ' W ↑G . Assume also that W
carries an H -invariant nonsingular symplectic F-form. Then there exists
a submodule Z of V ↓H , which also carries an H -invariant nonsingular
symplectic F-form, such that V ' Z↑G and | EndFG (V ) : EndFH (Z)| is an
odd number. If W is not induced from the normal core of H in G, then W
itself satisfies these conditions.
Proof. This is a consequence of 5.3, 5.14 and 5.18 (recall also that, since
W induces V from H , and it carries an H -invariant nonsingular symplectic F-form, then certainly V carries a G-invariant nonsingular symplectic
F-form (see 4.17)).
Then, going backwards through the chain of reductions obtained in
Chapters 3 and 4, also the weak versions of Conjecture 3.3 and Conjecture 3.2 can be proved to be true (with the relevant extra assumption). We
provide a few lines of proof for each of the following statements, just to recall
the path.
5.20. Theorem. Let G be a solvable group, H a subgroup of G having odd
prime index, F a finite field, V a simple FG-module, and W a submodule
of V ↓H such that V ' W ↑G . Assume also that V carries a G-invariant
nonsingular symplectic F-form f , which does not vanish on W . Then V
is form-induced (with respect to f ) from H . If W is not induced from the
normal core of H in G, then a submodule of V ↓H which form-induces V
can be chosen isomorphic to W .
Proof. This follows immediately by the previous theorem, together with
Lemma 4.22 and the discussion of paragraph 4.20; the latter, together with
Proposition 4.16, covers the case in which F has characteristic 2.
Observe that the last claim of 5.20 holds without any conditions on W
(concerning induction from the normal core), provided τV is not the identity,
or the characteristic of F is 2.
Finally, we go back to our original problem.
SOME POSITIVE ANSWERS TO THE CONJECTURES
85
5.21. Theorem. Let G be a solvable group, H a subgroup of G having odd
prime index, and D a faithful, primitive, tensor-indecomposable representation of G. Assume that we have D̄↓H ' P¯1 ⊗ P¯2 , where P1 and P2 are
projective representations of H . If deg P2 is not 1, and (deg P2 )|G:H| is
a divisor of deg D , then we have (deg P2 )|G:H| = deg D , and D is tensorinduced from H .
Proof. First of all, we go through Lemma 3.14; then, applying the
previous result and Theorem 3.13, we get the desired conclusion.
Having in mind another application of these results, still in the direction
of our conjectures, it is convenient to generalize theorem 5.18 dropping the
hypothesis of maximality for the subgroup H .
5.22. Theorem. Let G be a solvable group, H a subgroup of G having odd
index, F a finite field, V a simple FG-module which carries a G-invariant
nonsingular symplectic F-form f , and W a submodule of V ↓H . If V is
induced by W from H , and W is induced from the normal core L of H in
G, then V is form-induced (with respect to f ) from H .
Proof. We proceed by induction on the index of the subgroup. If H
is a maximal subgroup of G, then the claim is proved by Theorem 5.18;
therefore we can assume that there exists a proper subgroup M in G such
that H is properly contained in M . Now, V is induced by W from H ,
so we get V ' (W ↑M )↑G ; denoting by R the FM -module W ↑M , we have
that V is induced by R from M , and R is in turn induced from a normal
subgroup of G contained in M (which is L). We conclude that R is induced
from the normal core of M in G and, since |G : M | is odd, we can apply
the inductive hypothesis (we can certainly assume that R is a submodule of
V ↓M ) and find a submodule Y of V ↓M such that f does not vanish on Y ,
V ' Y ↑G , and | EndFG (V ) : EndFM (Y )| is an odd number. Next, We know
that there exists a submodule S of V ↓L such that V ' S↑G ; by Mackey’s
Lemma we get
M g M
V ↓M '
[(S )↑ ]
g∈T
where T is a set of representatives for the double cosets in G of L and M .
Since each of the (S g )↑M induces V from M and is therefore simple, we
have that Y is isomorphic, as an FM -module, to one of those. We conclude
that Y is induced from L, hence also from the normal core of H in M ;
thus we can use again the inductive hypothesis, obtaining that there exists
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5. POSITIVE RESULTS
a submodule Z of Y ↓H such that f ↓Y does not vanish on Z , Y ' Z↑M ,
and | EndFM (Y ) : EndFH (Z)| is an odd number. Now, putting together the
two steps, we see that Z is a submodule of V ↓H , f does not vanish on
Z , V ' Y ↑G ' (Z ↑M )↑G ' Z ↑G , and | EndFG (V ) : EndFH (Z)|, being the
product of | EndFG (V ) : EndFM (Y )| and | EndFM (Y ) : EndFH (Z)|, is an
odd number.
Of course one could ask what can be said, in the setting of Theorem
5.22, about form-induction of V from L; we shall give an answer to such a
question in Chapter 6, showing that in general we can not expect that V is
form-induced from L even if W is form-induced from L.
As an immediate consequence of 5.22, we also have the following results.
5.23. Theorem. Let G be a solvable group, H a normal subgroup of G
having odd index, and F a finite field; let V be a simple FG-module which
carries a G-invariant nonsingular symplectic F-form f , and W a submodule of V ↓H . If V is induced by W from H , then V is form-induced (with
respect to f ) from H .
5.24. Theorem. Let G be a solvable group, H a normal subgroup of G
having odd index, and D a faithful, primitive, tensor-indecomposable representation of G. Assume that we have D̄↓H ' P¯1 ⊗ P¯2 , where P1 and P2
are projective representations of H . If deg P2 is not 1, and (deg P2 )|G:H| is
a divisor of deg D , then we have (deg P2 )|G:H| = deg D , and D is tensorinduced from H .
Observe that, as we mentioned in the preamble of this chapter, Theorem 5.24 and Example 3.18 provide a full answer to the problem expressed
by Conjecture 3.2 if we restrict our attention to normal subgroups. in fact,
we are in a position to conclude that Conjecture 3.2 is true for normal subgroups of odd index, whereas it fails in general when the index of H is even.
As regards subgroups which are not normal, we have so far a positive answer
in the odd prime index case; if we could extend such a conclusion to the case
of subgroups having index an odd prime power, then some inductive argument, as in Theorem 5.22, could provide the general solution (recall that the
index of a maximal subgroup of a solvable group is always a prime power).
A final remark
In this chapter, and the previous one, we tended to work with modules over finite fields, essentially because this is the relevant setting for the
A FINAL REMARK
87
problems which we are studying. At any rate, it may be useful to observe
that in most cases we can generalize the results, replacing the hypothesis
of finiteness with the weaker hypothesis that the relevant field has prime
characteristic. This can be easily done by means of the following lemma (a
proof of it is provided in [GK], sec. 7).
5.25. Lemma. Let G be a group of order n, and F a field of prime characteristic; also, let F(n) be the (finite) subfield of the algebraic closure of
F generated by the n-th roots of 1, and let F0 denote the field F ∩ F(n) .
Then, for any subgroup H of G and for any simple FH -module V , there
exists a simple F0 H -module V0 , whose isomorphism type is uniquely determined, such that V ' V0F . Moreover, the endomorphism ring of V (which
is EndF0 H (V0 ) ⊗ F) is a field.
In order to get the desired generalizations, it is useful to take in account
Proposition 5.9 as well.
CHAPTER 6
An example
Looking at the discussion that we carried out so far, there are at least
three questions which arise naturally, and which really deserve an answer.
1. Consider the statement of Lemma 4.22.
Let G be a group, H a subgroup of G, F a field of odd characteristic,
V a simple FG-module, and W a submodule of V ↓H such that V ' W↑G .
Assume that W carries H -invariant nonsingular symplectic F-forms; then
the following conditions are equivalent:
(a) the automorphism τV is the identity on EndFG (V ), and the degree of
EndFG (V ) over EndFH (W ) (as a field extension) is an even number;
(b) there exists a G-invariant nonsingular symplectic F-form on V which
is not equivalent to any of the forms induced from W .
Moreover, for any G-invariant nonsingular symplectic F-form on V , there
exists a submodule of V ↓H , isomorphic to W , on which the relevant form
does not vanish.
Since we are interested in what happens when the index of H in G is
odd, one could ask whether such a condition is sufficient to avoid the ‘bad’
situation described in (a). After all, if H is a normal subgroup of odd index,
it follows immediately by Clifford’s Theorem that the homogeneous component of W in V ↓H has odd composition length, so that (by Lemma 5.3)
| EndFG (V ) : EndFH (W )| is an odd number. We shall see in a moment that,
for subgroups which are not normal, the answer to the question above is in
general ‘no’, since we have an example in which H has index 3, but condition (a) holds. This tells us that the relationship between induction and
form induction can be rather awkward, and not only when an ‘even step’ is
involved (as we have seen in Example 3.18).
The next question is very deeply related to the previous one.
2. Consider the statement of Theorem 5.14.
Let G be a solvable group, H a subgroup of G having odd prime index,
F a finite field of odd characteristic, V a simple FG-module, and W a
89
90
6. AN EXAMPLE
submodule of V ↓H such that V ' W ↑G . If W is not induced from the
L
normal core L of H in G, then we have V ↓H ' ( si=1 W ) ⊕ Y , where s is
an odd number and Y is a submodule of V ↓H such that HomFH (W, T ) = 0.
Can one omit the requirement that W is not induced from the normal
core of H ? Example 6.2 shows that the answer is ‘no’.
3. Consider the statement of Theorem 5.22.
Let G be a solvable group, H a subgroup of G having odd index, F a
finite field, V a simple FG-module which carries a G-invariant nonsingular
symplectic F-form f , and W a submodule of V ↓H . If V is induced by W
from H , and W is induced from the normal core L of H in G, then V is
form-induced (with respect to f ) from H .
Since V is induced from the normal core1 of H , we may ask whether
V is form-induced from L. The answer is in general ‘no’, and this is not
unexpected, since after all L may have even index in G.
We go now through an example which clarifies all these points. Before
that, it is useful to prove an easy general lemma.
6.1. Lemma. Let G be a group, H a subgroup of G, F a field, V an
FG-module, and W a submodule of V ↓H such that V ' W↑G ; assume also
that V ↓H is semisimple. If W is a simple homogeneous component of V ↓H ,
then V is simple.
Proof. Let Z be a submodule of V ; if Z contains W , then it also
contains all the translates of W by elements of G, whence it is all of V ; if
Z does not contain W , then Z lies in the sum of the other homogeneous
components of V ↓H ; that sum is the unique complement for W in V ↓H ,
therefore it coincides with the sum of all the translates of W (except for
W itself) by means of the elements of G. Our claim follows now from the
fact that the only subspace of that sum which admits the action of G is
obviously the zero space.
6.2. Example. We define the group G as A4 oQ16 ; the two groups involved
are generated as follows:
2
A4 = (C2 × C2 )oC3 = ha, ab , bi
1It sounds certainly strange to mention W in such a statement; we do it for coherence
with all the other theorems of this kind.
6. AN EXAMPLE
91
2
(here a, ab are generators of C2 × C2 , and b is a generator of C3 ), and
Q16 = hc, d : c4 = d2 = m , m2 = 1 , cd = c−1 i;
2
2
2
the action of Q16 on A4 is defined by ac = aab , (ab )c = ab , bc = b2 (the
action of d is trivial). Let us consider the subgroups
2
H := ha, ab , c, di ' (C2 × C2 )oQ16 ,
2
and L := ha, ab , c2 , di ' C2 × C2 × Q8 , which is the normal core of H in
2
G. We see that L/ha, ab c4 i is isomorphic to Q8 ; therefore, denoting by F
the prime field in characteristic 3, it is possible to define a 2-dimensional
FL-module Y , on which the elements c2 and d (whose cosets modulo
2
2
ha, ab c4 i generate L/ha, ab c4 i) act respectively as the matrices
!
!
0 −1
1
1
and
.
1
0
1 −1
We consider now the module W := Y ↑H , so that W = Y ⊕ Y c ; the
2
action of the four generators a, ab , c, d on W is given by the matrices

 
 



1
-1
1

, 
-1
1
-1
1
, 
-1
-1
-1
-1
 and 
1
1
1
-1
1
-1
1
-1
-1
,
and it is routine to check that W is an absolutely simple FH -module.
Next, we apply induction on W in order to construct an FG-module,
obtaining
2
2
2
V := W↑G = W ⊕ W b ⊕ W b = (Y ⊕ Y c ) ⊕ (Y b ⊕ Y cb ) ⊕ (Y b ⊕ Y cb ).
As V ↓H is a semisimple FH -module, we can get the decomposition of it
into its simple constituents, that is
2
2
V ↓H = (Y ⊕ Y c ) ⊕ (Y b ⊕ Y cb ) ⊕ (Y cb ⊕ Y b ).
2
The FH -module Y b ⊕ Y cb is acted upon by H exactly as W was, except
for the fact that a acts now as


-1
-1

.
1
1
92
6. AN EXAMPLE
At any rate, the first two simple constituents of V ↓H are isomorphic via the
matrix


-1
-1

1
-1
1
-1
1
1
.
2
Finally, on the FH -module T := Y cb ⊕ Y b the action of the four generators
of H is defined respectively by the matrices

 

 


-1
1
-1
-1

, 
-1
-1
1
, 
1
1
1
1
1
, and  -1
1
-1
-1
1
1
1
-1
.
In this case, we are dealing with an FH -module which is definitely not
isomorphic to the previous two; first of all it is not absolutely simple, as it
is possible to check that



y −y

 x
2
EndFH (Y cb ⊕ Y b ) =  y xy yx y  : x, y ∈ F .


−y
y
x
Moreover, looking at the decomposition of V ↓L into homogeneous components, we find
2
2
V ↓L = (Y ⊕ Y cb ) ⊕ (Y c ⊕ Y b ) ⊕ (Y cb ⊕ Y b )
and therefore T↓L is homogeneous, whereas W↓L is not.
We are now in a position to draw some conclusions. By the previous
lemma, V is a simple FG-module, since it is induced by T which is a
simple homogeneous component of V ↓H ; moreover, Y is endowed with an
L-invariant nonsingular symplectic F-form (it is clear that the action of L
on Y is given by two matrices which lie in Sp(2, 3)), so that W carries
an H -invariant nonsingular symplectic F-form. Since W is an absolutely
simple submodule of V ↓H , and its multiplicity in V ↓H is 2, Lemma 5.3
yields that EndFG (V ) has even degree (namely 2) over EndFH (W ) as a
field extension. Finally, any simple module for G (over an arbitrary field)
turns out to be self-contragredient, since every element of G is conjugate
to its inverse, and now Theorem 4.13 ensures that τV is the identity on
EndFG (V ). Such a situation provides an answer to Questions 1 and 2 (W
was constructed by induction from the normal core L of H in G).
Also, by Lemma 4.22 and paragraph 4.20, it is possible to find a
G-invariant nonsingular symplectic F-form g on V such that V is not
form-induced (with respect to g ) by any of the submodules of V ↓H which
6. AN EXAMPLE
93
are isomorphic to W (and g can not vanish on all of them); looking at the
statement of Theorem 5.20, we have that if W is induced from the normal core, in general we can not hope to find a submodule of V ↓H which
form-induces V and which is isomorphic to W . Of course now, by the same
Theorem 5.20, V has to be form-induced with respect to g by T . Indeed it
is; moreover, again by Lemma 4.22, V is form-induced by T with respect
to any G-invariant nonsingular symplectic F-form.
We have concluded that V is form-induced, with respect to g , by T from
H , and we know that V is induced from L; our aim is to show that V is not
form-induced, with respect to g , from L (this will answer to Question 3).
First of all, let us denote by f the form on V which arises by inducing the
L-invariant nonsingular symplectic F-form on Y mentioned above; of course
f↓T is an H -invariant nonsingular symplectic F-form on T . Moreover, we
2
have (Y cb )c = Y b and, with respect to this form, T ↓L is the orthogonal
2
direct sum of Y cb and Y b ; in other words, T is form-induced with respect
to f↓T from L. Assume now that V is form-induced from L with respect
to g ; then we get
V ↓L = Z y1 ⊥Z y2 ⊥ . . . ⊥Z y6
(here orthogonality is with respect to g ) where {y1 , . . . , y6 } is a transversal
for L in G, and Z is a submodule of V ↓L on which g does not vanish,
and such that V ' Z ↑G . Since T is a homogeneous component of V ↓L ,
then we have T = Z yi ⊥Z yj for some i, j . Moreover, T admits the action
of H , hence we get (Z yi )c = Z yj (in fact, Z yj c has to be Z yr for some
r in {1, . . . , 6}, and therefore it is Z yi or Z yj , but if it were Z yi then
yi cyi−1 would lie in L, so that c would lie in L as well, a contradiction).
We conclude that T is form-induced, with respect to the form g↓T , from
L. Now, let γ be the element of AutFG (V ) such that g(v1 , v2 ) = f (v1 , v2 γ)
for all v1 , v2 in V (see 4.4); as f and g are clearly inequivalent forms,
by 4.16(b) we know that γ is not a square in EndFG (V ). Of course we
get g(t1 , t2 ) = f (t1 , t2 γ↓T ) for all t1 , t2 in T ; if γ↓T , which is an element
of EndFH (T ), were equal to δ 2 for some δ in EndFH (T ), we would have
γ = δ 2 = δ̄ 2 (here δ̄ is the unique extension of δ to an FG-endomorphism of
V ), and therefore g(v1 , v2 ) = f (v1 , v2 δ̄ 2 ) for all v1 , v2 in V , a contradiction.
We conclude that f↓T and g↓T are inequivalent forms on T , and both of
them are equivalent to forms which are induced from L (because T is forminduced from L with respect to both of them). This leads to a contradiction:
in fact, by paragraph 4.18, τT is the identity (T induces V , and τV is the
94
6. AN EXAMPLE
identity), and by Lemma 5.3 the degree of EndFH (T ) over EndFL (Y cb ) is
even; now, by Lemma 4.22, we can not have two inequivalent H -invariant
nonsingular symplectic F-forms on T which are both induced from L.
APPENDIX
Counting anisotropic submodules
This appendix describes an earlier and rather different approach to the
problems discussed in the previous chapters, concerning in particular Conjecture 3.3. First of all, it is convenient to rephrase that conjecture in a
slightly different language, which suits the present context (recall Definitions 3.8 and 3.9).
Let G be a solvable group, H a subgroup of G, F a finite field, and V
a symplectic simple FG-module (it is clear that V ↓H is also a symplectic
module with respect to the relevant form on V ). Assume that there exists
an anisotropic simple submodule of V ↓H which induces V from H . Then
V is form-induced from H .
In our main approach to this problem we considered the whole set of
G-invariant nonsingular symplectic F-forms on V , rather than concentrating only on the given form1; then, after a series of steps in which the
concepts of induced form and equivalence of forms played a central role, we
obtained a reduction to Conjecture 4.1. In what follows, more simply (but
certainly less effectively), we focus on the problem in its original formulation, seeking directly a submodule of V ↓H which form-induces V from H .
Roughly speaking, we could say that in Chapter 4 and 5 we were ‘moving’ forms, whereas now we shall be moving submodules, the form being
fixed. In this case, we follow a rather ‘computational’ line; more precisely,
we consider a symplectic module which is semisimple and homogeneous and,
assuming that it contains an anisotropic simple submodule, we carry out a
counting of the anisotropic simple submodules of it. After such a computation, we shall be in a position to give an alternative proof of Conjecture
3.3 in the case that H is a normal subgroup of odd index (this is done in
Theorem A.13, which should be compared with Theorem 5.23). As we shall
1For this reason, so far it was often convenient to speak of ‘ FG -modules which carry
G -invariant nonsingular symplectic forms’ rather than of ‘symplectic modules’, as the
latter point of view suggests the presence of a well distinguished form on the relevant
module.
95
96
COUNTING ANISOTROPIC SUBMODULES
see, some of the fundamental aspects discussed in Chapters 4 and 5, like the
concept of equivalence of forms (‘hidden’ in A.1(c)) and the key role of the
field automorphism τ , are already recognizable in this earlier approach.
In the sequel, we always denote by h , i the given form on a symplectic
module, and τ stands for the field automorphism defined in Lemma A.1
(note that it is exactly the one described in Chapter 4); sometimes a subscript will refer τ to the relevant module. Also, the informations provided
by Lemma 3.16, concerning the number of simple submodules contained in
a semisimple module, will be useful in this context.
We begin with a lemma which defines an equivalence relation on the
set of simple submodules of a symplectic module (See [Is2], proof of Theorem 4.2). The role played by τ in characterizing the behaviour of such an
equivalence relation (see A.1(c,d)) reveals immediately the analogy of the
present approach with the main one developed in Chapters 4 and 5.
A.1. Lemma. Let H be a group, F a finite field, V a symplectic
FH -module which is semisimple and homogeneous, and W an anisotropic
simple submodule of V ; also, let E be the endomorphism ring EndFH (W ),
and let q denote the order of E.
(a) There exists a field automorphism τ of E, whose order is at most 2,
such that hxε, yi = hx, yετ i holds for all x, y in W and ε in E.
(b) If U and S are simple submodules of V , we say that U is equivalent
to S (and we write U ∼ S ) if there exists ϕ 6= 0 in HomFH (U, S)
such that hxϕ, yϕi = hx, yi for all x, y in U ; this yields an equivalence
relation on the set of simple submodules of V , and the isotropic simple
submodules form a single ∼-equivalence class.
(c) If U is a simple submodule of V , and ϑ is an FH -isomorphism from
W to U , then there exists a unique δ in E such that δ τ = δ and
hxϑ, yϑi = hx, yδi holds for all x, y in W . Moreover, we have U ∼ W
if and only if δ 6= 0 and δ = εετ for some ε in E.
(d) If τ is not the identity on E, then the anisotropic simple submodules of
V form a single ∼-equivalence class; if τ is the identity on E, then the
set of anisotropic simple submodules of V is partitioned into at most
two ∼-equivalence classes.
(e) Let U be an anisotropic simple submodule of V which is orthogonal to
W , and suppose that τ is the identity on E; suppose also that F has
odd characteristic. If −1 is a square in E× , then W ⊥U contains: two
isotropic simple submodules and two ∼-classes, containing (q − 1)/2
COUNTING ANISOTROPIC SUBMODULES
97
elements each, of anisotropic simple submodules in the case W ∼ U ;
no isotropic simple submodules and two ∼-classes, containing (q + 1)/2
elements each, of anisotropic simple submodules in the case W 6∼ U . If
−1 is not a square in E× , then the two cases above are switched.
(f) Let U be an anisotropic simple submodule of V which is orthogonal to
W , and suppose that τ is the identity on E; if the characteristic of F
is 2, then W ⊥U contains a unique isotropic simple submodule, and q
anisotropic simple submodules constituting a single ∼-equivalence class.
Proof of (a). The restriction of the form h , i to W × W is an
H -invariant nonsingular symplectic F-form on W , hence the claim follows
from 4.3, 4.6, and 4.9.
Proof of (b). This is clear.
Proof of (c). If the submodule U is anisotropic, then the map
f : W × W → F, (x, y) 7→ hxϑ, yϑi
is an H -invariant nonsingular symplectic F-form on W , and the first claim
of (c) follows from 4.3, 4.4 and 4.7. If U is isotropic, then of course the zero
endomorphism of W (and only that one) satisfies the required conditions.
Observe that, if ϑ0 is another FH -isomorphism from W to U , then
there exists ε in E× such that ϑ0 = εϑ, and the element of E attached to
ϑ0 as above is εδετ ; now the second claim of (c) can be easily proved.
Proof of (d). If τ is not the identity, it is clear that we have a single
∼-equivalence class of anisotropic simple submodules of V , as any element
of E× which is fixed by τ is of the kind εετ for some ε in E× . Assume
now that τ is the identity, and let W , Y , Y 0 be anisotropic simple submodules of V such that Y and Y 0 are both not equivalent to W . If ϑ is
an FH -isomorphism from W to Y , and ϑ0 an FH -isomorphism from W
to Y 0 , then the elements δ and δ 0 , attached (as in (c)) to the pairs (Y, ϑ)
and (Y 0 , ϑ0 ) respectively, are both non-squares in E× ; hence there exists ε
in E× such that δ = ε2 δ 0 . Now, consider the FH -isomorphism ϕ from Y
to Y 0 defined as the composite map ϑ−1 εϑ0 ; we get
h(xϑ)ϕ, (yϑ)ϕi = hxεϑ0 , yεϑ0 i = hx, yε2 δ 0 i = hx, yδi = hxϑ, yϑi
for all x, y in W . We conclude that Y and Y 0 are equivalent, and the
claim is proved.
Proof of (e). Suppose that −1 is a square in E× ; if we fix ϑ 6= 0 in
HomFH (W, U ), then every simple submodule X of W ⊥U , except for U ,
98
COUNTING ANISOTROPIC SUBMODULES
can be written as {w + wεϑ , w ∈ W } for some ε in E. Consider the map
γ : W → X defined as wγ := w + wεϑ; γ is clearly an FH -isomorphism,
and we have
hw1 γ, w2 γi = hw1 + w1 εϑ, w2 + w2 εϑi = hw1 , w2 i + hw1 , w2 ε2 δi
where δ is the element of E defined in (c). Now, X is isotropic if and only
if ε2 δ = −1, and this equation has two solutions provided we have W ∼ U
(in this case δ is a square in E× as well), otherwise it has no solutions.
Moreover, we have X ∼ W if and only if 1 + ε2 δ is a square in E× ; if δ is
a square, we find (q − 3)/2 elements ε in E such that 1 + ε2 δ is a square
different from 0 (see [Is2], Lemma 4.4) and, if we add the contribution of U ,
we get (q − 1)/2 anisotropic simple submodules in W ⊥U which are equivalent to W . It is now clear that the other ∼-equivalence class of anisotropic
simple submodules contains (q − 1)/2 elements as well. If δ is not a square,
then clearly we have X 6∼ X ⊥ for each anisotropic simple submodule X of
W ⊥U (here X ⊥ is meant to be the orthogonal of X in W ⊥U ); otherwise
W ⊥U = X⊥X ⊥ would contain two isotropic simple submodules, which is
not the case. Therefore, the map X 7→ X ⊥ provides a bijection between the
two ∼-equivalence classes of anisotropic simple submodules, whence they
contain (q + 1)/2 elements each.
If −1 is not a square in E× , the result can be obtained in a similar way.
Proof of (f). This is easily achieved, repeating the first few lines of the
previous point.
A.2. Remark. Let X be an anisotropic simple submodule of V , let EX
denote EndFH (X) and τX the field automorphism on EX defined in part
(a) of the lemma above; by part (d) we can see that, if τ is not the identity
on E, then τX is not the identity on EX . Indeed, let ε be an element of
E with ετ 6= ε, and let ϑ be an FH -isomorphism from W to X which
‘respects’ the form: for all x1 , x2 in X we have
hx1 ϑ−1 εϑ, x2 i = hx1 ϑ−1 ε, x2 ϑ−1 i = hx1 ϑ−1 , x2 ϑ−1 ετ i = hx1 , x2 ϑ−1 ετ ϑi,
whence (ϑ−1 εϑ)τX = ϑ−1 ετ ϑ holds, and it is clear that τX does not fix the
element ϑ−1 εϑ of EX .
The aim of the following lemmas is to compute the number of anisotropic
simple submodules of a symplectic module V , assuming that there is at
least one such submodule in it, and that V is semisimple and homogeneous.
Also in this situation the role of τ is crucial. Indeed, denoting by W an
COUNTING ANISOTROPIC SUBMODULES
99
anisotropic simple submodule of V , in Lemma A.7 we get an answer, concerning the number we are looking for, in the case that τ is the identity,
whereas in Lemma A.8 we get a different answer assuming that τ is an
involution.
A.3. Lemma. Let H be a group, F a finite field of odd characteristic, and
V a symplectic FH -module which is semisimple and homogeneous. Suppose
that V has an anisotropic simple submodule W which satisfies the following
conditions:
(a) τ is the identity on E := EndFH (W );
(b) −1 is a square in E× ;
(c) V = W ⊥W2 ⊥ . . . ⊥Wm , where the Wj are anisotropic simple submodules of V , all ∼-equivalent to W .
Also, let q denote the order of E. If the composition length m of V is odd
(say, m = 2k + 1), then V contains q 2k anisotropic simple submodules,
(q 2k + q k )/2 of them equivalent to W , (q 2k − q k )/2 of them constituting the
other ∼-equivalence class.
Proof. Consider the FH -module W2 ⊥W3 ⊥ . . . ⊥W2k+1 , that is W ⊥ ; if
X is a simple submodule of V which is different from W , denoting by π
the projection on W ⊥ , clearly Xπ is a simple submodule of W ⊥ and X is
contained in W ⊥ Xπ ; moreover, if R is a simple submodule of W ⊥ such
that X is contained in W ⊥R, then we have R = Xπ . It is now clear that,
denoting by SY the set of all simple submodules of an FH -module Y , we
have
[
(1)
SV =
(SW ⊥R \ {W }) ∪ {W }.
R∈S
W
⊥
We can now prove the lemma by induction on k . If k = 1 we have
V = W ⊥W2 ⊥W3 , and by Lemma A.1(e) we know that W ⊥W2 contains
two isotropic simple submodules, (q − 1)/2 anisotropic simple submodules
equivalent to W , and (q − 1)/2 anisotropic simple submodules not equivalent to W . If R ≤ W ⊥W2 is an isotropic simple submodule, then it is
easy to check that all the simple submodules of R⊥W3 are anisotropic and
equivalent to W (except of course for R). By Lemma A.1(e) and (1) we
conclude that the number of isotropic simple submodules of V is given by
q−1
q−1
2+0
+2
= q + 1,
2
2
100
COUNTING ANISOTROPIC SUBMODULES
hence V contains q 2 anisotropic simple submodules. Moreover, the number
of anisotropic simple submodules which are equivalent to W is given by
q−1 q−1
q−1 q+1
q2 + q
1 + 2(q − 1) +
−1 +
−1 =
2
2
2
2
2
and, of course, the other ∼-equivalence class contains
q2 −
q2 + q
q2 − q
=
2
2
anisotropic simple submodules.
Suppose now k greater than 1, and define S := W ⊥W2 ⊥ . . . ⊥W2k ,
T := W ⊥W2 ⊥ . . . W2k−1 . We can apply the inductive hypothesis on T , and
hence assume that T has q 2(k−1) anisotropic simple submodules, among
which (q 2(k−1) + q k−1 )/2 are equivalent to W , and (q 2(k−1) − q k−1 )/2 are
not equivalent to W . Therefore we can compute the number of isotropic
simple submodules of S , which is
2k−2
2k−3
X j
X j
q 2(k−1) + q k−1
=
q + q k−1 .
q +2
2
j=0
j=0
We can also compute the number of anisotropic simple submodules of S
which are equivalent to W ; this is
2k−3
X j
q 2(k−1) + q k−1 q − 1
1+
q (q − 1) +
−1
2
2
j=0
q 2(k−1) − q k−1
+
2
q+1
−1
2
q 2k−1 − q k−1
=
2
and the other ∼-equivalence class of anisotropic simple submodules contains
(q 2k−1 − q k−1 )/2 elements as well.
Finally, we can analyse the situation for V . The number of isotropic
simple submodules is given by
2k−2
2k−1
X j
X j
q 2k−1 − q k−1
k−1
q +q
+2
q ,
=
2
j=0
2k
j=0
hence we have q anisotropic simple submodules in V , as desired. The
number of simple submodules equivalent to W is now
2k−2
X j
q 2k−1 − q k−1 q − 1
k−1
1+
q +q
(q − 1) +
−1
2
2
j=0
q 2k−1 − q k−1 q + 1
q 2k + q k
+
−1 =
2
2
2
COUNTING ANISOTROPIC SUBMODULES
101
whereas the number of anisotropic simple submodules not equivalent to W
is of course (q 2k − q k )/2.
A.4. Lemma. Let H be a group, F a finite field of odd characteristic, and
V a symplectic FH -module which is semisimple and homogeneous. Suppose
that V has an anisotropic simple submodule W which satisfies the following
conditions:
(a) τ is the identity on E := EndFH (W );
(b) −1 is not a square in E× ;
(c) V = W ⊥W2 ⊥ . . . ⊥Wm , where the Wj are anisotropic simple submodules of V , all ∼-equivalent to W .
Also, let q denote the order of E. If the composition length m of V is odd
(say, m = 2k + 1), then V contains q 2k anisotropic simple submodules,
(q 2k + (−1)k q k )/2 of them ∼-equivalent to W , (q 2k + (−1)k+1 q k )/2 of them
constituting the other ∼-equivalence class.
We do not go into the details of the proof, as it is entirely similar to the
previous one.
We move now to characteristic 2.
A.5. Lemma. Let H be a group, F a finite field of characteristic 2, and
V a symplectic FH -module which is semisimple and homogeneous; suppose
that V has an anisotropic simple submodule W which satisfies the following
conditions:
(a) τ is the identity on E := EndFH (W );
(b) V = W ⊥W2 ⊥ . . . ⊥Wm , where the Wj are anisotropic simple submodules of V .
Then V has |E|m−1 anisotropic simple submodules.
Proof. The claim is easily proved by induction on m, recalling (1) in
Lemma A.3 and Lemma A.1(f).
A.6. Lemma. Let H be a group, F a finite field, and V a symplectic
FH -module which is semisimple and homogeneous. If V has an anisotropic
simple submodule W , then V admits an orthogonal direct decomposition in
which the summands are (anisotropic) simple submodules.
Proof. We proceed by induction on the composition length m of V .
For m = 2 the result is easily proved, since we have V = W ⊥W ⊥ and
W ⊥ is certainly an anisotropic simple submodule. Suppose then m > 2.
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COUNTING ANISOTROPIC SUBMODULES
In this case we also have V = W ⊥W ⊥ and, if W ⊥ has an anisotropic
simple submodule, then the result is achieved by means of the inductive
hypothesis. Therefore we may assume that all the simple submodules of W ⊥
are isotropic. In this case it is easy to check that, if S is a simple submodule
of V which is not contained in W ⊥ , then S is anisotropic: indeed, S can
be written as {w + wα , w ∈ W } for some α in HomFH (W, W ⊥ ) and hence,
for all s1 , s2 in S , we get
hs1 , s2 i = hw1 + w1 α, w2 + w2 αi = hw1 , w2 i + hw1 α, w2 αi.
But W α is a simple submodule of W ⊥ (therefore it is isotropic) and we have
hs1 , s2 i = hw1 , w2 i. It is now clear that, since W is anisotropic, we can not
have hs1 , s2 i = 0 for all s1 , s2 in S . Choose now a simple submodule of
V , say R, such that R 6= W and R 6≤ W ⊥ (such a submodule certainly
exists, as HomFH (W, W ⊥ ) is not the zero space). We have R⊥ ∩ W ⊥ < R⊥ ,
otherwise R⊥ would lie in W ⊥ so that W would lie in R, which is not
the case; since R⊥ is semisimple, we get R⊥ = Z ⊕ (R⊥ ∩ W ⊥ ), where
Z is a simple (because of its dimension) submodule of V . Observe now
that R⊥ has a simple submodule, namely Z , which is anisotropic (as it is
not contained in W ⊥ ), and the result is achieved applying the inductive
hypothesis to R⊥ .
A.7. Lemma. Let H be a group, F a finite field, and V a symplectic
FH -module which is semisimple and homogeneous. If V has an anisotropic
simple submodule W such that τ is the identity on E := EndFH (W ), and
if the composition length m of V is odd, then V has |E|m−1 anisotropic
simple submodules.
Proof. By Lemma A.6 we can write V as W1 ⊥W2 ⊥ . . . ⊥Wm , where the
Wj are (anisotropic) simple submodules of V . We know by Remark A.2 that
τWJ is the identity on EWj ; moreover, it is easy to see that we can assume
Wj ∼ W1 for all j in {2, . . . , m} (this is possible because m is odd). The
result follows now from Lemmas A.3, A.4 and A.5.
We remark that an entirely similar computation (which gives a different
answer) can be carried out in the case that the composition length of V is
even. Now we move to the case in which τ is an involution.
A.8. Lemma. Let H be a group, F a finite field of characteristic p, and
V a symplectic FH -module which is semisimple and homogeneous. If V
has an anisotropic simple submodule W such that τ is not the identity on
COUNTING ANISOTROPIC SUBMODULES
103
E := EndFH (W ), then we have |E| = p2k for some k in N, and the number
of anisotropic simple submodules of V is given by
f (m) = hm−1 ·
hm + (−1)m+1
h+1
where h is pk and m is the composition length of V .
Proof. Let r be an integer such that |E| = pr ; we know that the group
of field automorphisms of E, Aut(E), is a cyclic group of order r , and it
contains an element τ of order 2. This forces r to be even, whence we get
r = 2k for some k in N.
Consider now any two anisotropic simple submodules of V , say X and
Y ; by Lemma A.1(d) we have X ∼ Y , therefore we can choose ϑ in
HomFH (X, Y ) such that hx1 ϑ, x2 ϑi = hx1 , x2 i for all x1 , x2 in X . If X
and Y are orthogonal, we can determine the number of isotropic simple
submodules contained in X⊥Y as follows: every simple submodule S of
X⊥Y , except for Y , can be written as {x + xεϑ , x ∈ X} where ε is a
suitable element of EX := EndFH (X); therefore we have
hs1 , s2 i = hx1 + x1 εϑ, x2 + x2 εϑi = hx1 , x2 i + hx1 , x2 εετX i
for all s1 , s2 in S , and it is now clear that S is isotropic if and only if
εετX = −1. Since τX is the (unique) element of order 2 in Aut(EX ) (see Rek
mark A.2), we get ετX = εp = εh , so that the equation becomes εh+1 = −1
and it has h + 1 solutions in EX . We conclude that X⊥Y contains h + 1
isotropic simple submodules, so that the number of anisotropic simple submodules of it is given by h2 + 1 − h − 1 = h2 − h.
Now we can prove the lemma by induction on m. The discussion above
yields the result for m = 2, so that we may assume m > 2. Observe
that, by Lemma A.6, we can also assume V = W ⊥W2 ⊥ . . . ⊥Wm , where
the Wj are anisotropic simple submodules of V ; moreover, the FH -module
W ⊥ = W2 ⊥W3 ⊥ . . . ⊥Wm satisfies the hypotheses of the lemma. We already
know that, if X and Y are simple submodules of V which are orthogonal,
then X⊥Y has f (2) = h2 − h anisotropic simple submodules in the case
that X and Y are both anisotropic, whereas it contains h2 anisotropic
simple submodules if X (or Y ) is anisotropic and the other one is isotropic.
Recalling (1) in Lemma A.3, we can compute the number of anisotropic
104
COUNTING ANISOTROPIC SUBMODULES
simple submodules of V , which is given by
f (m) = 1 + f (m − 1) · (h2 − h − 1)+
!
h2(m−1) − 1
+
− f (m − 1) · (h2 − 1) =
2
h −1
= f (m − 1) · (−h) + h2(m−1) .
Now we are in a position to apply the inductive hypothesis, and the desired
conclusion is achieved.
In order to prove Conjecture 3.3 for normal subgroups of odd index, we
need some more preparation.
A.9. Lemma. Let H be a group, F a finite field, and V a symplectic
FH -module which is semisimple and homogeneous. If V has an anisotropic
simple submodule X such that τX is not the identity on EX := EndFH (X)
then, for each anisotropic simple submodule W of V , there exist anisotropic
simple submodules W2 ,. . . ,Wm of V such that V = W ⊥W2 ⊥ . . . ⊥Wm .
Proof. We proceed by induction on the composition length m of V :
if m = 2 we have V = W ⊥W ⊥ , where W ⊥ is an anisotropic simple submodule of V . Then we assume m > 2, and we have V = W ⊥W ⊥ as
well; if W ⊥ contains an anisotropic simple submodule, we can apply the
inductive hypothesis on W ⊥ and we are done. Hence we suppose that
every simple submodule of W ⊥ is isotropic and, as we saw in Lemma A.6,
we conclude that every simple submodule of V which is not contained in
W ⊥ is anisotropic. Now we have two types of simple submodules in V : the
anisotropic simple submodules, and the simple submodules of W ⊥ ; therefore
the identity
h2m − 1
h2 − 1
=
h2(m−1) − 1
m−1
hm + (−1)m+1
·
h+1
+h
h2 − 1
h2m − 1
hm−1 m−1
= 2
+ 2
(h
+ (−1)m−1 h − hm + (−1)m )
h −1
h −1
holds, that is,
hm−1 + (−1)m−1 h = hm + (−1)m−1 .
(Here h is |EX |1/2 ). It is easy to see that such an identity is inconsistent,
and we get the contradiction which completes the proof.
COUNTING ANISOTROPIC SUBMODULES
105
A.10. Lemma. Let H be a group, F a finite field, and V a symplectic
FH -module which is semisimple and homogeneous. Suppose that V contains an anisotropic simple submodule W such that τ is not the identity on
E := EndFH (W ), and define the number
g(m) =
m
1 Y i−1 hi + (−1)i+1
·
,
h
·
m!
h+1
i=2
1/2
where h is |E|
and m denotes the composition length of V . Then g(m)
is the number of all the possible decompositions of V in the orthogonal direct
sum of anisotropic simple submodules.
Proof. This follows at once from the previous lemma.
We also recall a well known result.
A.11. Lemma. Let G be a group, H a normal subgroup of G, F a finite
field of characteristic p, and V a simple FG-module. If the order of the
quotient group G/H is p, and V ↓H is homogeneous, then V ↓H is simple.
(See [HB], VII, 9.19).
Now we can prove Conjecture 3.3 for normal subgroups of odd prime
index. As the last preliminary ingredient, we recall the following elementary
fact: let p be a prime, and G a p-group which acts (via a permutation
representation) on a finite set Ω; if p does not divide the cardinality of Ω,
then there are elements in Ω which are fixed by G.
A.12. Lemma. Let G be a group, H a normal subgroup of G whose index is an odd prime t, and F a finite field. If V is a symplectic simple
FG-module, and V ↓H has an anisotropic simple submodule W such that
t dim W = dim V , then there exist anisotropic simple submodules X1 , . . . , Xt
of V ↓H which satisfy the following properties:
(a) dim Xi = dim W for all i in {1, . . . , t};
(b) V ↓H = X1 ⊥X2 ⊥ . . . ⊥Xt ;
(c) G permutes the Xi in its action by translation on the set of simple
submodules of V ↓H .
Proof. First of all, observe that V ↓H is a symplectic semisimple
FH -module. Since G/H permutes transitively the homogeneous components of V ↓H , the number of them is necessarily 1 or t; in the latter case
W itself is a homogeneous component of V ↓H and, fixed a right transversal
106
COUNTING ANISOTROPIC SUBMODULES
{1, g2 , . . . , gt } for H in G, we get
V ↓H = W ⊕ W g2 ⊕ · · · ⊕ W gt
where each of the W gj is a homogeneous component of V ↓H as well. Since
W ⊥ is a complement for W in V ↓H , but the unique complement for W
is the sum of the other homogeneous components, we conclude that the
direct decomposition above is orthogonal (and clearly stabilized by G), as
requested.
We are left with the case that V ↓H is homogeneous. This condition,
together with the hypothesis that V ↓H has a proper submodule (namely
W ), forces t to be different from the characteristic of F (see Lemma A.11);
moreover, the composition length of V ↓H is t (an odd number). In such a
situation, we can conclude that τ is not the identity on E := EndFH (W ).
Indeed, if τ were the identity, V ↓H would have |E|t−1 anisotropic simple
submodules (by Lemma A.7), but G/H acts on the set of anisotropic simple
submodules of V ↓H and t = |G/H| is not a divisor of |E|t−1 ; this leads to
a contradiction, since G/H would be forced to stabilize some anisotropic
simple submodule X of V ↓H , and therefore X would be a proper submodule
of V .
We are now in a position to apply Lemma A.10, obtaining that the
number of orthogonal direct decomposition of V ↓H , in which the summands
are anisotropic simple submodules, is given by
g(t) =
t
1 Y i−1 hi + (−1)i+1
·
h
·
t!
h+1
i=2
1/2
where h is defined as |E| .
The next step is to show that t is not a divisor of g(t). Since G/H acts
on the set of anisotropic simple submodules of V ↓H , and the cardinality of
this set is given by f (t) = ht−1 (ht + 1)/(h + 1) (see Lemma A.8), t is forced
to be a divisor of f (t). This yields h ≡ −1(t), which in turn implies that
t2 is not a divisor of (ht + 1)/(h + 1). Indeed, since h can be written as
nt − 1 for some n in N, we see that (ht + 1)/(h + 1) is a polynomial in
P
Pt
t whose 1-degree term is ( t−1
i=1 −in)t and whose 0-degree term is
i=1 1
2
(these only are the terms which could be possibly not divisible by t ). Now
we have
X
t−1
−nt2 (t − 1)
,
−in t =
2
i=1
COUNTING ANISOTROPIC SUBMODULES
107
Pt
hence that term is actually divisible by t2 , while
i=1 1 is t; we conclude
t
2
that (h + 1)/(h + 1) ≡ t(t ) holds. Finally, since we have h ≡ −1(t), t is
not a divisor of (hi + (−1)i+1 )/(h + 1) for any i which is strictly smaller
than t. The conclusion is that t!g(t) is not divisible by t2 , so that g(t) is
not divisible by t, as claimed.
Now, G/H acts in a natural fashion on the set of decompositions of
V ↓H in the orthogonal direct sum of anisotropic simple submodules. As the
cardinality of this set is not divisible by t, we must have a fixed point, and
the proof is complete.
Finally, arguing by induction, we can drop the assumption that the index
of H in G is prime. What we get is a statement which express Conjecture 3.3
for normal subgroups of odd index, although it is formulated in a different
way.
A.13. Theorem. Let G be a solvable group, H a normal subgroup of G
whose index is an odd number m, and F a finite field. If V is a symplectic simple FG-module, and V ↓H has an anisotropic simple submodule W
such that m dim W = dim V , then there exist anisotropic simple submodules
X1 , . . . , Xm of V ↓H which satisfy the following properties:
(a) dim Xi = dim W for all i in {1, . . . , m};
(b) V ↓H = X1 ⊥X2 ⊥ . . . ⊥Xm ;
(c) G permutes the Xi in its action by translation on the set of simple
submodules of V ↓H .
Proof. We proceed by induction on m. If H is a maximal normal subgroup of G, then m is an odd prime and our claim follows from Lemma A.12.
Assume that there exists a normal subgroup Z of G with H < Z < G,
and choose a right transversal T1 := {z1 = 1, z2 , . . . , zs } for H in Z ,
and a right transversal T2 := {y1 = 1, y2 , . . . , yr } for Z in G. Then
T1 T2 := {zi yj , 1 ≤ i ≤ s , 1 ≤ j ≤ r} is a right transversal for H in
G. Since |G : H| dim W = dim V , we have
V ↓H =
r M
s
M
W zi yj
j=1 i=1
(which means V ' W↑G ); if we denote by R the FH -module W z1 ⊕· · ·⊕W zs
it is clear that R is a simple submodule of V ↓Z (indeed, we have V ' R↑G )
which is anisotropic; otherwise it would be isotropic, and this is not possible
because R contains W . Now we can apply the inductive hypothesis, since
108
COUNTING ANISOTROPIC SUBMODULES
|G : Z| is an odd number and |G : Z| dim R equals the dimension of V . We
conclude that there exists an anisotropic simple submodule Y of V ↓R with
dim Y = dim R and such that V ↓Z = Y y1 ⊥Y y2 ⊥ . . . ⊥Y yr .
We shall show that Y ↓H contains an anisotropic simple submodule S
with dim S = dim W . Indeed, let us denote by πj the projection on Y yj
for j in {1, . . . , r}; πj is a morphism of FH -modules, so that W πj is a
submodule of Y yj which is trivial or FH -isomorphic to W (in this case
of course it is simple with the same dimension as W ). Moreover, we have
certainly W ≤ W π1 ⊥W π2 ⊥ . . . ⊥W πr , hence it is clear that at least one of
the W πj , say W πk , has to be an anisotropic simple submodule of V ↓H and
(replacing Y with Y yk ) we get what we wanted.
Now we can apply the inductive hypothesis on the FZ -module Y with
respect to S , and we conclude that there exists an anisotropic simple submodule X of Y ↓H with dim X = dim S = dim W , and such that
Y ↓H = X z1 ⊥X z2 ⊥ . . . ⊥X zs .
Therefore we have
V ↓H = ⊥rj=1 (⊥si=1 X zi yj ),
and the proof is complete.
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