THE INVARIANT THEORY OF K−1 [X1 , X2 , . . . , XN ] UNDER PERMUTATION
REPRESENTATIONS
BY
J. CAMERON ATKINS
A Thesis Submitted to the Graduate Faculty of
WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES
in Partial Fulfillment of the Requirements
for the Degree of
MASTER OF ARTS
Mathematics
May 2012
Winston-Salem, North Carolina
Approved By:
Ellen Kirkman, Ph.D., Advisor
Sarah K. Mason, Ph.D., Chair
W. Frank Moore, Ph.D.
Acknowledgments
I would like to thank my parents for being so patient with me through my adolescents. When I decided to pursue a path in mathematics, they supported me all the
way, and they continue to support me as I head to the University of South Carolina.
My siblings Jared and Whitney treated me as the younger of the three, and over the
past years we have grown to truly love each other. I feel lucky to have the relationship
we share and encouragement they give me. I would also like to thank Melissa Bechard.
She and I began studying mathematics our freshmen year at James Madison University. We struggle and we fight through problems together, and if nothing else, I have
fun exploring math with her. Lastly, I would like to thank my advisor Dr. Kirkman.
Out of everyone she is the most deserving of my praise. As a professor, I have never
learned more from one individual. Her rigor from the classroom reappeared in her
guidance as an advisor, and I am a better mathematician for it.
ii
Table of Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Groups and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Group Representation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Invariant Theory of k[x1 , x2 . . . , xn ] Under Permutation Matrices . . .
1.4 The Non-Commutative Ring k−1 [x1 , x2 , · · · , xn ] . . . . . . . . . . . .
1
1
2
2
5
Chapter 2 Orbit Sums and the Symmetric Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Orbit Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
9
Chapter 3 Göbel’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Special Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Göbel’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 4 The Alternating Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Antisymmetric Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 RAn Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 5 Examples With Göbel’s Bound with 4 indeterminates. . . . . . . . . . . . . .
5.1 Alternating Z2 Subgroup . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Alternating Klein IV Subgroup . . . . . . . . . . . . . . . . . . . . .
5.3 Non-Alternating Klein IV Subgroup . . . . . . . . . . . . . . . . . . .
33
33
36
38
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Appendix A
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Appendix B
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Vita. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
iii
Abstract
Let k be a field of characteristic zero, and let R = k−1 [x1 , · · · , xn ] denote the skewpolynomial ring with xj xi = −xi xj for i 6= j. The symmetric group Sn acts on R
as permutations of the {xi }. Given a subgroup G of permutations in Sn we consider
the problem of finding algebra generators for RG , the subring of invariants under
G. We find generators for RG when G is the full symmetric group Sn and when G
is the alternating group An . We describe an algorithm that produces generators of
RG for a general subgroup G of Sn . The algorithm produces a an upper bound on
the degrees of algebra generators of RG , generalizing the Göbel bound for invariants
under permutation actions on k[x1 , · · · , xn ].
iv
Chapter 1:
1.1
Introduction
Groups and Rings
Let G be a group and S be a set. We define a left group action on S by G to be
G × S → S, where (g, s) = gs, and if e is the identity of G the action satisfies es = s
and g(hs) = (gh)s for all g, h ∈ G and for all s ∈ S. We call the set S a G-set. The
orbit of element s ∈ S under a group G is the set of elements t in S such that there
exists some g ∈ G with gs = t. We denote the orbit of an element s ∈ S under a
group G to be OrbG (s). A stabilizer of element s ∈ S is an element of g ∈ G such
that gs = s. The set of all stabilizers of an element s ∈ S from G forms a subgroup
of G denoted StabG (s). We let Sn denote the symmetric group on n elements and
An be the alternating group.
A ring is a set R with two binary operations + and · which are called addition
and muliplication respectively. The ring under addition forms an abelian group. The
ring under multiplication satisfies the left and right distribution laws, i.e. a(b + c) =
ab + ac and (a + b)c = ac + bc, as well as being associative, i.e. a(bc) = (ab)c. A
field is a commutative ring with unity where every non-zero element is a unit. The
characteristic of a field, k, is the smallest positive integer n such that ∀r ∈ k, nr = 0.
If no positive integer exists we say the ring has characteristic zero. Throughout this
thesis k is always a field of characteristic zero. We can construct polynomials with
n-indeterminates with coefficients from a field k denoted k[x1 , x2 , . . . , xn ] or R. Let
I = (i1 , i2 , . . . , in ) be a vector of length n of non-negative integers, and let X I denote
the monomial xi11 xi22 · · · xinn .
1
1.2
Group Representation
A linear representation of a group G is a homomorphism ϕ : G → Gl(n, k), where k
is any field, and Gl(n, k) is the set of n × n invertible matrices with entries from k.
A representation, ϕ, is called faithful if the kernel of ϕ is only the identity of G. The
−1 0
example ϕ : Z4 → Gl(2, R), with ϕ(1) =
defines a representation fo Z4 ,
0 1
where ϕ(j) = ϕ(1 + 1 + · · · + 1) (one is added to itself j times). This representation
is not fiathful to since ker(ϕ) = {0, 2}. On the other hand, ρ : Z3 → Gl(2, C)
2πi
e 3 0
where ρ(1) =
defines a faithful representation since ker(ρ) = {0}. A
0 1
permutation matrix is denoted as Mσ , and is the matrix where the entries mi,j = 1
if σ(j) = i and mi,j =0 otherwise. For example consider S4 and the permutation
σ = (1, 2, 3). Then the permutation matrix is

0
 1
Mσ = 
 0
0
0
0
1
0
1
0
0
0

0
0 
.
0 
1
The representation of the symmetric group Sn with φ : Sn → Gl(n, R) where φ(σ) =
Mσ is called a permutation representation of Sn . This representation is faithful. In
this thesis we explore the permuation representation of the symmetric group as well
as the subgroups of the symmetric group under the same representation.
1.3
Invariant Theory of k[x1 , x2 . . . , xn ] Under Permutation Matrices
For any subgroup G of Sn , represented as permutation matrices, we define a G-action
on the set of monomials. If σ is a permutation of {1, 2, . . . , n} then we define
i
1
2
n
σ(xi11 xi22 · · · xinn ) = xiσ(1)
xiσ(2)
· · · xiσ(n)
= x1σ
2
−1 (1)
i
x2σ
−1 (2)
i
· · · xnσ
−1 (n)
.
This action is extended to R = k[x1 , · · · , xn ] by
σ
X
aI X I =
X
I
aI σ(X I ).
I
Given any subgroup G of Sn and an element f ∈ R, we say f is invariant under G
if ∀g ∈ G, g(f ) = f . The set of elements of R that are invariant under G forms a
subring of R that we denote by RG . This thesis concerns finding algebra generators
for the invariant subring RG , i.e. finding elements f1 , · · · , fm of RG such that any
invariant element f ∈ RG can be written as sum of the form
f=
X
im
) for aI ∈ k.
aI (f1i1 f2i2 · · · fm
I=(i1 ,i2 ,...,im )
One way to produce invariants is to form orbit sums. For any monomial X I , let
OG (X I ) denote the sum of the elements in the G-orbit of X I , namely
OG (X I ) =
X
g(X I ).
OrbG (X I )
Any orbit sum will be an invariant in RG . Moreover, it can be shown that any element
of RG is a linear combination of orbit sums [2]. When G is the full symmetric group
Sn an orbit sum can be represented by its leading term under the lexicographic order
with x1 > x2 > · · · > xn ; the exponent sequence of that leading term will be weakly
decreasing and hence form a partition into n parts of the sum of the exponents. Hence,
orbit sums under Sn correspond with partitions into n parts. When G = Sn Gauss
proved that the n elementary symmetric polynomials are a generating set for RSn ,
where the elementary symmetric polynomials and the corresponding partitions are:
e1 (x1 , . . . , xn ) = OSn (x1 ) = x1 + x2 + · · · + xn ↔ (1, 0, 0, · · · 0),
e2 (x1 , . . . , xn ) = OSn (x1 x2 ) =
X
i6=j
3
xi xj ↔ (1, 1, 0, · · · 0),
X
e3 (x1 , . . . , xn ) = OSn (x1 x2 x3 ) =
xi xj xk ↔ (1, 1, 1, 0, · · · 0),
i,j,k distinct
..
.
en (x1 , . . . , xn ) = OSn (x1 x2 · · · xn ) = x1 x2 · · · xn ↔ (1, 1, 1, · · · 1).
Power sums form another algebra generating set for the symmetric functions RSn ,
where for 1 ≤ i ≤ n the n power sums are defined by
pi = OSn (xi1 ) = xi1 + xi2 + · · · + xin ↔ (i, 0, 0, · · · 0).
Newton proved the Newton Symmetric Formulas, which are relations between the
power sums and the elementary symmetric polynomials, and Waring gave a formula
expressing the power sums in terms of the elementary symmetric polynomials [2]. It
has also been shown that either set of algebra invariants is algebraically independent
(i.e. the invariant subring is a polynomial ring). Hence, invariants under the full
symmetric group can be written uniquely in terms of the elementary symmetric polynomials or the power sums [3]. Under other subgroups of the symmetric group, the
invariant subring is generally not a polynomial ring. In other words, for a particular set of generators of an invariant subring, RG , polynomials in RG are not written
uniquely in terms of the generators.
The Vandermonde Determinant ∇n = Πi>j (xi − xj ) is invariant under An but not
Sn . It has been shown that the elementary symmetric polynomials along with the
Vandermonde Determinant are algebra generators of RAn [3].
More recently the problem of describing algebra generators for RG , where G is a subgroup of Sn under the permutation representation, was considered. In his 1996 thesis
Manfred Göbel determined a set of algebra generators for RG and an upper bound
for their degrees, now called the Göbel bound. Göbel’s generating set is usually not
4
a minimal generating set, but the Göbel bound can be sharp. The Göbel bound is
very useful because it provides the largest degree of orbit sums one needs to consider
in finding an algebra generating set for RG . Göbel’s generating set consists of the
elementary symmetric polynomials along with orbit sums of called “special monomials.” An exponent sequence, I, is called special if rewriting I in a weakly decreasing
order denoted λ(I) where λ(I) = (λ1 , λ2 , . . . , λn ) satisfies the following:
1) λi − λi+1 ≤ 1
2) λn = 0.
n
The Gob̈el bound is the maximum of n and
. Since the Vandermonde Determi2
n
nant has degree
, the Göbel bound is sharp for G = An for n ≥ 3 [1].
2
1.4
The Non-Commutative Ring k−1 [x1 , x2 , · · · , xn ]
We define the ring R = k−1 [x1 , x2 , · · · , xn ] to be the ring over a field k of characteristic
zero with n indeterminates with the property xi xj = −xj xi when i 6= j. For example
consider the ring k−1 [x1 , x2 , x3 ] and the element x2 x1 − 3x1 x23 + 4x23 x32 = −x1 x2 −
nm m n
xj xi . The elements of this ring are called
3x1 x23 + 4x32 x23 . In general xni xm
j = (−1)
”skew-polynomials” instead of polynomials since these elements have the property
xi xj = −xj xi when i 6= j. We note that permutations of the indeterminates xi
extend to automorphisms of R = k−1 [x1 , x2 , · · · , xn ] because any transposition of
{1, 2, · · · , n} preserves the relations xi xj = −xj xi . For a fixed scalar q ∈ k there are
other noncommutative algebras kq [x1 , x2 , · · · , xn ], with defining relations xi xj = qxj xi
for i > j. However, the only q for which all transpositions preserve these relations
are when q = 1 (the classical commutative case) and q = −1 (the case considered
5
here). This thesis provides results for k−1 [x1 , x2 , . . . , x2 ] that are analogous to those
described for k[x1 , x2 , . . . , xn ] in Section 1.3.
6
Chapter 2:
Orbit Sums and the Symmetric Group
Let k be a field of characteristic zero, and adjoin a finite number of indeterminates with the property xi xj = −xj xi when i 6= j. We denote this ring as
R = k−1 [x1 , x2 , . . . , xn ]. Let G be a subgroup of a permutation group acting faithfully on the ring with g(c) = c, for any c in k and g(xi ) = xj . We extend g to an
automorphism of k−1 [x1 , . . . , xn ] by defining g on monomials by
g(xi11 · · · xinn ) = g(x1 )i1 · · · g(xn )in ,
and defining g on the skew-polynomials ring by
g
X
aI X I =
X
I
aI g(X I ).
I
Unlike the commutative case, we note that in the skew-polynomial case to rewrite
g(X I ) with the variables in ascending order may require interchanging variables, and
hence may introduce sign changes. For example, x53 x72 = −x72 x53 . If f in R satisfies
g(f ) = f for all g in G, then we say f is invariant under G. These invariants form a
subring denoted RG .
2.1
Orbit Sums
In this section we produce orbit sums as invariants, and we show that any invariant
is a linear combination of orbit sums. The lemma below reduces the search for invariants to the search for homogeneous invariants, i.e. skew-polynomials where all the
terms have the same total degree.
Lemma 2.1.1. Homogeneous components of invariants under a linear action are
invariants themselves.
7
Proof. Since g does not change the total degree of a monomial, homogeneus compoP
nents of f are invariant. That is, if f = m
i=1 fni where the terms of fni all have the
same degree, then g(f ) = f implies g(fni ) = fni .
Definition 1. An orbit sum of a monomial X I under the action of a group G is the
sum of the distinct monomials in the orbit of X I under G. An orbit sum is denoted
OG (X I ).
Note that
X
OG (X I ) =
g(X I ) 6=
OrbG (X I )
X
g(X I ).
g∈G
If we sum over every element in the group some terms in the orbit sum may have a
higher multiplicity than one.
Example 2.1.2. Consider the ring k−1 [x1 , x2 , x3 ]S3 and the monomials x71 x32 and
x21 x22 x3 . The orbit sum of x71 x32 is
OS3 (x71 x32 ) = x71 x32 + x71 x33 − x31 x72 − x31 x73 + x72 x33 − x32 x73 .
The orbit sum of x21 x22 x3 is
OS3 (x21 x22 x3 ) = x21 x22 x3 + x21 x2 x23 + x1 x22 x23 .
Note the orbit sum of a monomial under a group action G is invariant under G.
Theorem 2.1.3. Every invariant is a linear combination of orbit sums.
Proof. Let f be in RG , and by Lemma 2.1.1 we can assume without loss of generality
that f is homogeneous. Let
f=
X
aI X I with aI 6= 0.
I
8
We want to show that f is a linear combination of orbit sums; i.e.
f=
X
aJ OG (X J ).
J
Let’s consider some element g ∈ G acting on f . Since f is invariant, we have g(f ) = f .
If we look at g acting on some particular summand we see
g(aI X I ) = g(aI )g(X I ) = aI g(X I ),
and
i
n
2
1
= ±x1g
...xig(n)
xig(2)
g(X I ) = g(xi11 xi22 ...xinn ) = xig(1)
−1 (1)
i
x2g
−1 (2)
i
...xng
−1 (n)
.
The plus or minus sign comes from the possible sign change to put variables into
P
ascending order. Since f is invariant g(X I ) is a term in I ai X I , and since distinct
P
powers are linearly independent the coefficient of g(X I ) in I aI X I must be ag(I) .
Hence aI = ag(I) . Thus all monomials in the orbit of X I will have coefficient aI , and
f=
X
aJ OG (X J )
J
is the sum of orbit sums.
2.2
Symmetric Group
In this section we describe two families of polynomials that generate RSn as an algebra. These families are analogous to the power sums and the elementary symmetric polynomials in the commutative case that generate subring of invariants of
k[x1 , x2 , . . . , xn ] under the permutation action of the full symmetric group.
In the non-commutative ring k−1 [x1 , x2 , . . . , xn ], unlike the commutative case, some
orbit sums are equal to zero.
9
Example 2.2.1. Consider k−1 [x1 , x2 ] and G = S2 .
OS2 (x1 x2 ) = x1 x2 + x2 x1
= x1 x2 − x1 x2
= 0.
This raises the question of when an orbit sums is zero. Later in this thesis we
define algorithms that reduce the exponent sequence of particular monomials. Thus
we have to be careful that we do not reduce an orbit sum to the zero polynomial.
Since the set of monomials of degree d are linearly independent over the vector space
of all polynomials of degree d, if O(X I ) = 0 then for any term X J occuring in orbit
of X I , the monomial −X J must also occur in the orbit.
Note that G ⊆ Sn acts on a monomial by
i
1
2
n
g(X I ) = g(xi11 xi22 · · · xinn ) = xig(1)
xig(2)
· · · xig(n)
= ±x1g
−1 (1)
i
x2g
−1 (2)
i
· · · xng
−1 (n)
,
so when g acts on the monomial and the variables are arranged in ascending order,
the exponent sequence goes from I to g −1 (I). Also OG (X I ) is a sum of X to powers
that are all g −1 (I) for g ∈ G. If G = Sn then the exponent sequences are all the
permutations of I. Hence it is convenient to represent the orbit sums by OSn (X I )
where I is a partition. Thus each orbit sum over Sn corresponds to a partition of
Pn
I
j=1 ij into n parts for I = {ij }. Futhermore, X (where I is a partition) is the
leading term under the lexicographic order, where x1 > x2 > · · · > xn , of the orbit
sum.
Lemma 2.2.2. An orbit sum, OSn (X I ), is zero if and only if there are at least two
repeated odd entries in I.
Proof. ⇐) Since I is a partition, the pair of repeated odd entries can be assumed to
be adjacent. Let ij = ij+1 be odd. Consider the transposition (j, j + 1) acting on X I .
10
i
i
j
(j, j + 1)X I = (j, j + 1)(xi11 xi22 ...xjj xj+1
...xinn )
i
i
j
xjj ...xinn
= xi11 xi22 ...xj+1
i
i
j
= −xi11 xi22 ...xjj xj+1
...xinn
= −X I .
Decompose the group Sn into a union of distinct left cosets of the subgroup
H = h(j, j + 1)i, so
Sn =
[
σH.
σ∈Sn
The orbit of X I under H is {X I , −X I }. The image of X I under the two elements in
the coset, σH, are σ(X I ) and −σ(X I ). For any element, X J , in the orbit of X I ,its
negative, −X J , is also in the orbit of X I ; therefore, the orbit sum is zero.
⇒) Let the orbit sum OSn (X I ) = 0. Since the orbit sum is zero each term needs
to pair with its negative. If (−X I ) is to occur in the orbit, then some nonidentity
element σ ∈ Sn must be an element of Stab(I) and σ(X I ) = −(X I ). Therefore,
there must be at least one pair of repeated entries in I. Assume I = (i1 , i2 , ..., in )
has no repeated odd entries, so every σ from Stab(I) acting on I must fix every odd
entry. Thus, every σ in Stab(I) acting on (X I ) is a permutation of the even powered
indeterminates. Thus, for any σ from Stab(I), σ(X I ) = X I . So −X I cannot occur in
the orbit, hence the orbit sum cannot be zero. Therefore, there must be a repeated
pair of odd entries in the exponent sequence for the orbit sum to be zero. Hence,
OSn (X I ) = 0 if and only if there are two repeated odd entries.
Lemma 2.2.3. The largest monomial under the lexicographic order in the product of
two non-zero orbits sums, OSn (X I ) and OSn (X J ), has degree I + J, where I and J
are partitions. Also the orbit sum OSn (X I+J ) 6= 0.
11
Proof. Assume OSn (X I ) 6= 0 and OSn (X J ) 6= 0. (X I )(X J ) = ±X I+J is a term in
OSn (X I )OSn (X J ). If you consider any other term, σ(X I ) × τ (X J ), either I or J
will be permuted or both will be permuted. A permutation of a partition (P) gives
an exponent sequence that is less than or equal to (P) under the lexicographic order.
Since we are dealing with orbit sums, each term may only appear once, implying
that both exponent sequences may not be fixed by the permutations. So one of the
exponents, σ(I) or τ (J) must be lower lexicographically than I or J respectively. Thus
σ(I) + τ (J) <lex I + J. Hence, all terms in OSn (X I )OSn (X J )other than ±X I+J must
be lower lexicographically. The monomial ±X I+J occurs once in OSn (X I )OSn (X J )
and all other summands are lower lexicographically, implies that ∓X I+J cannot occur.
Hence, OSn (X I+J ) 6= 0.
Now we introduce two sets of invariants: the n odd power sums and the n elementary skew-symmetric polynomials.
Definition 2. The n odd power sums are P := {OSn (xi1 )|1 ≤ i ≤ 2n − 1 and i is
odd}. We denote a particular polynomial from P as pi where i determines the degree.
Definition 3. The n elementary skew-symmetric polynomials are OSn (X I ) where I
can be any of the following sequences: (1,0,...,0),(2,1,0,...,0), (2,2,1,0,...,0),...,(2,2,...,2,1).
We denote a particular skew-polynomial from the set as Sa = OSn (X I ), where a denotes the position where the 1 is in the sequence.
Example 2.2.4. Consider R = k−1 [x1 , x2 , x3 ]. The 3 odd power sums for R are p1 =
OS3 (x1 ) = x1 +x2 +x3 , p3 = OS3 (x31 ) = x31 +x32 +x33 , and p5 = OS3 (x51 ) = x51 +x52 +x53 .
The 3 elementary skew-symmetric polynomials are OS3 (x1 ) = x1 +x2 +x3 , OS3 (x21 x2 ) =
x21 x2 + x21 x3 + x1 x22 + x22 x3 + x1 x23 + x2 x23 , and OS3 (x21 x22 x3 ) = x21 x22 x3 + x21 x2 x23 + x1 x22 x23 .
12
The n odd power sums are the analog to the n power sums, {OSn (xi )|1 ≤ i ≤ n},
in the commutative case. The n elementary skew-symmetric polynomials are the
analog to the elementary symmetric polynomials in the commutative case.
Lemma 2.2.5. k[x21 , x22 , ..., x2n ]Sn is contained in the algebra generated by P and also
in the algebra generated by the n elementary skew-symmetric polynomials.
Proof. Since x2i is in the center of R, k[x21 , x22 , ..., x2n ]Sn is commutative and
k[x21 , x22 , ..., x2n ]Sn ' k[y1 , y2 , ..., yn ]Sn where the set {yi , y2 , . . . , yn } is algebraically
independent. This ring k[y1 , y2 , ..., yn ]Sn is generated as an algebra either by the
power sums or the n elementary symmetric polynomials [3]. Therefore, the ring
k[x21 , x22 , ..., x2n ]Sn is generated by {OSn (xi1 ) where 2 ≤ i ≤ 2n and i is even} or
{OSn (x21 x22 ...x2a )|1 ≤ a ≤ n}.
Since k[x21 , x22 , ..., x2n ]Sn is contained in k−1 [x1 , x2 , · · · , xn ]Sn , we show that it is
2
2
2 Sn
generated by P. Let OSn (x2m
.
1 ) where 1 ≤ m ≤ n be a generator for k[x1 , x2 , ..., xn ]
We claim that 12 (p1 p2m−1 + p2m−1 p1 ) = OSn (x2m
1 ). First we compute:
n
X
p1 p2m−1 = (x1 + ... + xn )(x2m−1
+ ... + xn2m−1 ) =
1
xi xj2m−1 .
1≤i,j≤n
Consider the j
th
term in
n
X
xi x2m−1
when we arrange the variables in ascending
j
1≤i,j≤n
order.
If i = j then the term is x2m
i .
If i < j then the term is xi xj2m−1 .
If i > j then the term is xi xj2m−1 = −x2m−1
xi .
j
On the other hand,
P2m−1 P1 =
(x2m−1
1
+ ... +
x2m−1
)(x1
n
+ ... + xn ) =
n
X
1≤j,i≤n
13
x2m−1
xi .
j
Consider the jith term in
n
X
x2m−1
xi .
j
1≤j,i≤n
If i = j then the term is x2m
i .
If i > j then the term is x2m−1
xi .
j
If i < j then the term is x2m−1
xi = −xi x2m−1
.
j
j
!
n
n
X
X
2m
2m
xi x2m−1
+
x2m−1
xi = x2m
So 12
1 + ... + xn = OSn (x1 ).
j
j
1≤i,j≤n
1≤j,i≤n
Hence, k[x21 , x22 , . . . , x2n ]Sn is contained in the algebra generated by P. Now we
show k[x21 , x22 , · · · , x2n ]Sn is contained in the algebra generated by the n elementary
skew-symmetric polynomials.
Let OSn (X I ) = OSn (x21 x22 . . . x2a ) be a generator of k[x21 , x22 , ..., x2n ].Sn We claim
that OSn (X I ) =
1
(S1 Sa
2a
+ Sa S1 ).
First we want to show S1 Sa = OSn (x31 x2 x23 . . . x2a−1 xa ) + aOSn (x21 x22 . . . x2a ). We are
considering Sn -invariants, so we know that every Sn orbit sum is defined by monomials with exponent sequences that are partitions. Thus we consider every partition that can be constructed from permutations of (1, 0, · · · , 0) summed with permutations of (2, 2, · · · , 2, 1, 0, · · · , 0). There are only 3 such partitions, which are
(3, 2, · · · , 2, 1, 0, · · · , 0), (2, · · · , 2, 0, · · · , 0), and (2, · · · , 2, 1, 1, 0, · · · , 0). The partition (2, · · · , 2, 1, 1, 0, · · · , 0) has a repeated odd entry, thus by Lemma 2.2.2 this orbit
sum is trivial. Now we must determine the multiplicity as well as the sign of the other
orbit sums. The partition (3, 2, · · · , 2, 1, 0, · · · , 0) is the largest lexicographically in
terms of S1 Sa , so by the proof of Lemma 2.2.3 it occurs once, and its sign is positive,
i.e. (x1 )(x21 · · · x2a−1 xa ) = x31 x22 . . . x2a−1 xa . The final partition (2, · · · , 2, 0, · · · , 0) occurs when (x1 )(x1 x22 · · · x2a ), (x2 )(x21 x2 x23 · · · x2a ), . . . ,
(xa )(x21 · · · x2a−1 xa ). Since the first term xi only commutes past variables with even exponents, the sign of the monomial is always positive. Thus the partition (2, · · · , 2, 0, · · · , 0)
14
occurs a times. Hence, S1 Sa = OSn (x31 x2 x23 . . . x2a−1 xa ) + aOSn (x21 x22 . . . x2a ). Similarly, Sa S1 produces the same partitions with the same multiplicity, but the partition (3, 2, · · · , 2, 1, 0, · · · , 0) will be negative since x1 must commute past exactly
one variable with an odd power. Therefore, Sa S1 = −OSn (x31 x22 x23 . . . x2a−1 xa ) +
aO.Sn (x21 x22 . . . x2a ). Thus,
1
(S1 Sa
2a
+ Sa S1 ) =
2a
O (x21 x22
2a Sn
. . . x2a ) = OSn (x21 x22 . . . x2a ).
Hence, k−1 [x21 , x22 , . . . , x2n ]Sn ⊆ k−1 [x1 , x2 , . . . , xn ]Sn is contained in the algebra
generated by either P or the elementary skew-symmetric polynomials.
Theorem 2.2.6. The ring k−1 [x1 , x2 , ..., xn ]Sn is generated as an algebra by the n
elementary skew-symmetric polynomials, {Sa }.
Proof. We proceed by induction on the lexicographic order of the leading monomial of
an invariant. Let f be an invariant with leading monomial of degree zero. Hence f is
an element of the field, so it can be written as a trivial polynomial in the elementary
skew-symmetric polynomials. Let cX I be the leading monomial of f where I is a
partition and is not of degree zero. We will subtract products of elementary skewsymmetric polynomials and invariants of lower leading length lexicographic order from
f to remove the highest lexicographic term. Assume there are no odd entries in I.
According to Lemma 2.2.5, OSn (X I ) can be written as a linear combination of powers
of elementary skew-symmetric polynomials. So the leading monomial in f − OSn (X I )
is of lower degree than f . Assume I = (i1 , ..., in ) has at least one odd entry. Let
ia be the last odd entry in the sequence. Since I is a partition and OSn (X I ) is not
∗
trivial, we have ij ≥ ia + 1 ≥ 2. for all ij , where j < a. Consider OSn (X I )Sa where
I ∗ = (i1 − 2, i2 − 2, ..., ia−1 − 2, ia − 1, ia+1 , ..., in ), which is also a partition of non∗
negative entries. By Lemma 2.2.3, the largest monomial in OSn (X I )Sa is X I and all
∗
other terms are lower lexicographically than X I . So all terms of f − cOSn (X I )Sa ,
for some c ∈ k, are lower lexicographically than f . Since the degree is finite and there
are only a finite number of exponent sequences of lower lexicographic order than I,
15
this process terminates. Hence, any skew-polynomial f ∈ RSn can be written as
elementary skew-symmetric polynomials.
Example 2.2.7. Consider k−1 [x1 , x2 , x3 ]S3 and the skew-polynomial OS3 (x41 x2 ) =
x41 x2 + x41 x3 + x1 x42 + x42 x3 + x1 x43 + x2 x43 . In the exponent sequence I = (4, 1, 0), the
last odd entry is the second entry. So we subtract two from all the higher entries and
one from second entry, and we get I ∗ = (2, 0, 0). So we multiply OS3 (x21 )OS3 (x21 x2 ) =
x41 x2 + x41 x3 + x1 x42 + x42 x3 + x1 x43 + x2 x43 + x31 x22 + x31 x23 + x21 x32 + x21 x33 + x32 x23 + x22 x33 +
2x21 x22 x3 + 2x21 x2 x22 + 2x1 x22 x23 = OS3 (x41 x2 ) + OS3 (x31 x22 ) + 2OS3 (x21 x22 x3 ). In this
step we get an orbit OS3 (x31 x22 ) that we also need to reduce. So we continue the
process and we consider OS3 (x21 x22 )OS3 (x1 ) = x31 x22 + x31 x23 + x21 x32 + x21 x33 + x22 x33 +
x32 x23 + x21 x22 x3 + x21 x2 x22 + x1 x22 x23 = OS3 (x31 x22 ) + OS3 (x21 x22 x3 ). Hence, OS3 (x41 x2 ) =
OS3 (x21 )OS3 (x21 x2 ) − OS3 (x21 x22 )OS3 (x1 ) − OS3 (x21 x22 x3 ).
Lastly we need to write
OS3 (x21 x22 ) in terms of the elementary skew-symmetric polynomials. From Lemma
2.2.5, OS3 (x21 x22 ) = 12 (OS3 (x1 )OS3 (x21 x2 )+OS3 (x21 x2 )OS3 (x1 )). Therefore, OS3 (x41 x2 ) =
OS3 (x1 )2 OS3 (x21 x2 )− 12 OS3 (x21 x2 )OS3 (x1 )2 − 12 OS3 (x1 )OS3 (x21 x2 )OSn (x1 )−OS3 (x21 x22 x3 ).
We will show that P also generates k−1 [x1 , x2 , . . . , xn ]Sn , but we need to introduce
a new order on the set of degree d monomials. In the commutative case under the
usual lexicographical order on monomials we have an order on orbit sums determined
by the leading term. In degree d ≤ n the orbit sum with the largest leading term is the
power polynomial Pd = OSn (xd1 ), and the orbit sum with the smallest leading term is
the elementary symmetric polynomial Sd = OSn (x1 x2 · · · xd ). Sturmfels defined the
order below and used it to prove the power sums generate the invariant subring when
Sn acts on the commutative polynomial ring k[x1 , · · · , xn ] [3]. Next we define this
order and adapt his proof to the noncommutative ring k−1 [x1 , · · · , xn ].
Definition 4. A monomial (xi11 xi22 . . . xinn ) ≺ (xj11 xj22 . . . xjnn ) if the partition
λ(i1 , i2 , . . . in ) >lex λ(j1 , j2 , . . . , jn ), or if λ(i1 , i2 , . . . in ) =lex λ(j1 , j2 , . . . , jn ).
16
This order forms a partial order on the set of monomials in R. Under ≺ the n elementary skew-symmetric polynomials are considered the largest. This is because for a
fixed degree d ≤ 2n − 1, the smallest degree d monomial in terms of the lexicographic
order, is created by assigning exactly one 1 and assigning other exponents twos or
zeros. If we assign more than one 1 then we create a trivial orbit sum. Similarly, the n
odd power sums are the smallest under ≺ since they are the largest lexicographically.
Theorem 2.2.8. The ring k−1 [x1 , x2 , . . . , xn ]Sn is generated by the n odd power sums,
P.
Proof. By Theorem 2.2.6 the n elementary skew-symmetric polynomials generate
RSn , so it suffices to show that they can be generated by the n odd power sums
P. Hence, it is enough to show that we can express invariant skew-polynomials of
total degree less than or equal to 2n − 1 in terms of P. Let f ∈ RSn with leading
P
term cX I = xi11 xi22 · · · xinn where nt=1 it ≤ 2n − 1. Let Pi = OSn (xi1 ), a power sum,
where i can be even or odd. Consider the invariant P = Pi1 Pi2 . . . Pin whose largest
monomial under is dxi11 xi22 . . . xinn . Let f := f − dc P . The largest monomial in f
must be less than X I , since X I was the largest in f . This process terminates since f
and f have the same degree. Thus the ring k−1 [x1 , x2 , . . . , xn ]Sn is generated by the
power sums. From Lemma 2.2.5, the power sum Pi where i is even can be expressed
by the n odd power sums. Hence, f can be expressed by the n odd power sums.
Example 2.2.9. Consider the ring k−1 [x1 , x2 , x3 , x4 ] and rewriting the elementary
skew-symmetric polynomial OS4 (x21 x22 x3 ) in terms of odd power sums. The first step
from the algorithm above is OS4 (x21 x22 x3 ) − 21 (OS4 (x21 ))2 OS4 (x1 ) = − 21 OS4 (x51 ) −
1
O (x4 x )
2 S4 1 2
− OS4 (x31 x22 ). Again, we need to rewrite OS4 (x41 x2 ) and OS4 (x31 x22 ) as
power sums, and we need to use Lemma 2.2.5 to write OS4 (x21 ) as odd power sums.
We continue the algorithm and see that the orbit sum OS4 (x41 x2 ) = OS4 (x41 )OS4 (x1 )−
17
OS4 (x51 ) and OS4 (x31 x22 ) = OS4 (x31 )OS4 (x21 ) − OS4 (x51 ). We continue the algorithm
further and see that OS4 (x21 x22 x3 ) = 12 (OS4 (x21 ))2 OS4 (x1 ) − 12 OS4 (x41 )OS4 (x1 )
− OS4 (x31 )OS4 (x21 ) + OS4 (x51 ). Then by Lemma 2.2.5 we see OS4 (x21 x22 x1 ) =
1
(OS4 (x1 )2 )2 OS4 (x1 )
2
− 14 (OS4 (x1 )OS4 (x31 ) + OS4 (x31 )OS4 (x1 )) OS4 (x1 ) −
OS4 (x31 )OS4 (x1 )2 + OS4 (x51 ).
Thus, OS4 (x21 x22 x3 ) can be written as odd power sums.
18
Chapter 3:
Göbel’s Bound
Now we want to determine a set of generators for RG where G is a subgroup of
Sn . If a skew-polynomial f from R is invariant under all elements of Sn , then f
is invariant under G; thus RSn ⊆ RG . We will use the elementary skew-symmetric
polynomials as part of our generating set
3.1
Special Monomials
Given a subgroup G of Sn we define a class of special monomials that will be used
to produce a generating set RG .
Definition 5. A monomial, X I , is called special if the exponent sequence I has an
associated partition λ(I) with:
i) λi (I) − λi+1 (I) ≤ 2 for all 1 ≤ i ≤ n, and
ii) λn (I) is either 1 or 0.
Example 3.1.1. Consider the ring k−1 [x1 , x2 , x3 , x4 , x5 ].
The sequences I = (1, 0, 3, 4, 2) and J = (2, 2, 1, 2, 0) are both special, where as the
sequences K = (1, 3, 1, 0, 0) and L = (5, 4, 3, 2, 1) are not special.
For every monomial X I we associate a special monomial Red(X I ) through an
iterative process generalized from Göbel’s algorithm for k[x1 , x2 . . . , xn ] [1]. When
λi (I) − λi+1 (I) > 2, we call this a gap in the exponent sequence. Then for all j ≤ i
subtract 2 from λj (I). Repeat this procedure until there are no more gaps. This
produces the special exponent sequence Red(I) and special monomial Red(X I ). X I
and Red(X I ) differ by a factor with even exponents and thus the factor is in the
19
center of R. We can write X I = Red(X I )X J where X J = X I /Red(X I ). Notice by
construction that the ith largest entry in Red(I) is in the same position the ith largest
entry in J. This implies the permutation λ that takes (I) to its associated partition
λ(I) has the property λ(I) = λ ((Red(I)) + (J)) = λ(Red(I)) + λ(J), where λ(J) and
λ(Red(I)) are both partitions of J and Red(I).
Example 3.1.2. Let R = k−1 [x1 , x2 , ..., x7 ], and consider X I where I = (0, 7, 4, 1, 0, 1, 7).
The first gap appears between 7 and 4. We reduce every entry that is greater than
or equal to 7 by 2. We get (0,5,4,1,0,1,5). There is still a gap between 4 and 1 so
we need to reduce again. We get (0,3,2,1,0,1,3). This new exponent sequence is now
special. Thus X I = x72 x43 x4 x6 x77 , Red(X I ) = x32 x23 x4 x6 x37 , and X J = x42 x23 x47 .
There is a concern that when reducing an exponent sequence I to Red(I) we can
get a trivial orbit sum, i.e. OG (Red(X I )) = 0.
Lemma 3.1.3. If OG (Red(X I )) = 0 then OG (X I ) = 0.
Proof. Let Red(X I ) be the associated special monomial of X I and X I = Red(X I )X J .
Assume OG (Red(X I )) = 0, so ∃σ ∈ G such that σ(Red(X I )) = −Red(X I ). Thus,
σ(Red(I)) = Red(I). Since σ(Red(I)) = Red(I), we have σ(J) = J. By construction,
J is an exponent sequence where every entry is even. Therefore σ acting on (X J )
cannot create a sign change. Hence, σ(X J ) = X J . Recall X I = Red(X I ))X J , and
consider:
σ(X I ) = σ(Red(X I )X J )
= σ(Red(X I ))σ(X J )
= −Red(X I )X J
= −X I
Since σ ∈ G, we have that −X I ∈ OrbG (X I ). Therefore, OG (X I ) = 0.
20
Lemma 3.1.4. For any monomial X I and for every σ ∈ Sn ,
(a) σ(Red(X I )) = Red(σ(X I ))
(b) σ(Red(X I )) = Red(X I ) if and only if σ(X I ) = X I .
Proof. (a) follows since permuting a sequence then reducing is the same as reducing
then permuting. (b) If σ(X I ) = X I then σ(Red(X I )) = Red(σ(X I )) = Red(X I ).
Conversely, if σ(Red(X I )) = Red(X I ) then Red(σ(X I )) = Red(X I ). Thus σ(X I ) =
XI.
3.2
Göbel’s Bound
We use the same order on the set of monomials that Derksen used to prove Göbel’s
theorem in the commutative case [1].
Definition 6. We say X I X J if the first difference, i, between the associated
partitions λ(I) and λ(J) is λi (I) > λi (J). We say X I X J if either the first
difference, i, between λ(I) and λ(J) is λi (I) < λi (J), or if λj (I) = λj (J), for all j.
Example 3.2.1. Let X I , X J , and X K be monomials with exponent sequences
I = (0, 5, 1, 2), J = (5, 0, 1, 1), and K = (2, 5, 0, 1). The partitions of the exponent sequences are λ(I) = (5, 2, 1, 0), λ(J) = (5, 1, 1, 0), and λ(K) = (5, 2, 1, 0). The
first difference in the partition of I and J is the second entry. λ2 (I) = 2, λ2 (J) = 1
so λ2 (I) > λ2 (J). Thus X I X J . Now X I X K since λj (I) = λj (K) for all j.
Theorem 3.2.2. Let X I be some monomial, and X J = X I /Red(X I ). Every monomial occurring in OSn (X J )OG (Red(X I )) − OG (X I ) is lower than X I under order.
To prove this theorem we will prove:
(a) X I X K , ∀X K occuring in OSn (X J )Red(X I ) − X I .
21
We consider this case where OSn (X J ) is multiplied by the monomial Red(X I ), since
it is less cumbersome than considering OSn (X I )OG (Red(X I )). Then we utilize part
(a) along with Lemma 3.1.3 to prove:
(b) X I X K , ∀X K occuring in OSn (X J )OG (Red(X I )) − OG (X I ).
Proof. (a) Clearly X I occurs in the sum, OSn (X J )Red(X I ). Let’s assume to the
contrary that there exists X K = σ(X J )Red(X I )such that X I X K , where σ ∈ Sn .
We want to show that σ(X J ) = X J ; then (a) is proved.
Let Red(X I ) = Πni=1 xei i and X J = Πni=1 xdi i . Thus X I and X K have exponents ei + di
and ei + dσ−1 (i) respectively. We reorder X I such that the exponent sequence is a
n
2
1
for some permutation τ . By construction,
· · · xλτ (n)
xλτ (2)
partition, i.e. X I = ±xλτ (1)
the permutation τ that takes I to its partition is the same permutation that takes
J and Red(I) to their respective partitions. So we may assume that ei ≥ ei+1 and
di ≥ di+1 . Let m be maximal with d1 = d2 = ... = dm . Since X I X K we know
that σ must permute the set {1, 2, ..., m}. Similarly, we consider the second biggest
exponent, and maximal l such that dm+1 = ... = dm+l . Since X I X K we know
that σ permutes {m + 1, m + 2, ..., m + l}. We continue this process and see that the
exponent sequence (d1 , ..., dn ) is invariant under σ. Thus σ(X J ) = X J .
(b) For any σ ∈ G we have the following as a consequence of 3.2.2(a),
I
σ(X )
I
I
σ(X I ) X K for all X K occurring in OSn ( Red(σ(X
I )) )Red(σ(X )) − σ(X ).
Thus X I X K . By Lemma 3.1.3(a),
I
σ(X )
I
J
I
J
I
OSn ( Red(σ(X
I )) )Red(σ(X )) = OSn (σ(X ))σ(Red(X )) = OSn (X )σ(Red(X )).
Therefore, X I X K for all X K occuring in OSn (X J )σ(Red(X I )) − σ(X I ).
22
By Lemma 3.1.3(b), the permutations σ ∈ G, such that σ(X I ) = X I are exactly
those that fix Red(X I ). So summing over the coset representation, σ, of the stabilizer
in G of X I , we obtain X I X K for all terms in OG (X J )OG (Red(X I ))−OG (X I ).
Example 3.2.3. Consider k−1 [x1 , x2 , x3 , x4 ]Z4 where < g >= Z4 , and

0
 1
g=
 0
0
0
0
1
0
0
0
0
1

1
0 
.
0 
0
Let I = (2, 0, 5, 1), and therefore, Red(I) = (2, 0, 3, 1) and J = (0, 0, 2, 0). So
OZ4 (Red(X I ))OS4 (x21 ) = OZ4 (x41 x33 x4 ) + OZ4 (x21 x22 x33 x4 ) + OZ4 (x21 x53 x4 ) + OZ4 (x21 x33 x34 ).
Therefore, OZ4 (Red(X I ))OS4 (x21 ) − OZ4 (X I ) = OZ4 (x41 x33 x4 ) + OZ4 (x21 x22 x33 x4 ) +
OZ4 (x21 x33 x34 ).
The remaining exponent sequences are (4, 3, 1, 0), (3, 2, 2, 1), (3, 3, 2, 0). All of them
are lower in comparison with I = (2, 0, 5, 1) under order.
Corollary 3.2.4. The degree of the largest possible generator k−1 [x1 , x2 , . . . , xn ]G is
n2 .
Proof. From Göbel’s algorithm a set of generators of RG is the orbit sums of special
monomials along with the n-elementary skew-symmetric polynomials. The largest
elementary skew-symmetric polynomial in terms of total degree is 2n-1. The largest
orbit sum of a special monomial, OG (X I ), in terms of total degree has the associated
partition λ(I) = (2n − 1, 2(n − 1) − 1, 2(n − 2) − 1, . . . , 5, 3, 1). Therefore, the degree
P
of the orbit sum is ni=1 (2i − 1) = n2 . The value n2 ≥ 2n − 1, ∀n ≥ 1. Thus, the
largest possible generator of RG has degree n2 .
We do not know if the bound is sharp.
23
Chapter 4:
The Alternating Group.
Now we want to determine a set of algebra generators for RAn . Since An ≤ Sn ,
we have RSn ⊆ RAn .
4.1
Antisymmetric Invariants
In this section we begin our consideration of invariants under the alternating group
An by discussing antisymmetric invariants. In the commutative case the set of antisymmetric invariants and the symmetric polynomials generate k[x1 , x2 , . . . , xn ]An [3].
Analogously, the set of antisymmetric skew-polynomials and symmetric skew-polynomials
generate RAn .
Lemma 4.1.1. An orbit sum OAn (X I ) is invariant under Sn if and only if there
exists some odd permutation τ such that τ (X I ) = X I .
Proof. (⇒) Let OAn (X I ) be invariant under Sn . Therefore OrbSn (X I ) = OrbAn (X I ),
and |StabSn (X I )| = 2|StabAn (X I )|. Thus, StabSn (X I ) must contain some permutation that StabAn (X I ) does not. Hence there is some odd permutation that stabilizes
XI.
(⇐). Since [StabSn (X I ) : StabAn (X I )] ≤ [Sn : An ] = 2, so if some odd permutation σ stabilizes X I , then |StabSn (X I )| = 2|StabAn (X I )|. Notice |Sn | = 2|An | =
2|OrbAn (X I )||StabAn (X I )| = |OrbSn (X I )||StabSn (X I )| = 2|OrbSn (X I )||StabAn (X I )|.
Therefore, OrbAn (X I ) = OrbSn (X I ), and OSn (X I ) = OAn (X I ).
Let f be an An -invariant and let σ and τ be any two odd permutations. Then
τ −1 σ is even, so τ −1 σ(f ) = f , and hence σ(f ) = τ (f ).
24
Lemma 4.1.2. The orbit sum OAn (X I ) = 0 if and only if there are exponents ij = ik ,
with ik odd, and is = it , with it even.
Proof. (⇒) Let OAn (X I ) = 0. Similarly to the Sn case, we need some permutation,
σ, from An such that σ(X I ) = −X I . If (−X I ) is to occur in the orbit, then some
nonidentity element σ ∈ An must be an element of Stab(I) and σ(X I ) = −(X I ). If
σ fixes all the odd exponents then −X I cannot occur. Thus σ must permute at least
one pair of exponents ij with ik where ij = ik , and ik is odd. Also since σ is an even
permutation, σ must permute at least one other pair of exponents and not create a
sign change. Thus, there must be some pair of exponents it = is , where it is even
which σ permutes. Therefore, σ permutes j with k and t with s.
(⇐) Let the exponent sequence I have some indices ij = ik , an odd number, and
is = it , an even number. Consider the even permutation (j, k)(s, t) acting on X I . So
(s, t) fixes X I since it permutes two even numbers. Then (j, k)(X I ) = −X i . Thus
the orbit sum OAn (X I ) = 0.
Corollary 4.1.3. The invariant OAn (Πn−1
i=1 xi ) is the smallest in terms of total degree
that is both an An -invariant and not a Sn -invariant.
n−1
Proof. The exponent sequence of OAn (Πi=1
xi ) has no repeated evens, thus by Lemma
n−1
/ RSn . Any other orbit sum, f,
4.1.1. and 4.1.2. OAn (Πn−1
i=1 xi ) 6= 0 and OAn (Πi=1 xi ) ∈
that is lower in terms of degree will have at least two zeros in the exponent sequence.
n−1
Thus by Lemma 4.1.1, f is a Sn -invariant. Hence OAn (Πi=1
xi ) is the smallest degree
An -invariant that is not a Sn -invariant.
Definition 7. A skew-polynomial f is called antisymmetric if τ f = −f for every
odd permutation in Sn .
Theorem 4.1.4. Every An -invariant decomposes uniquely into a sum of a symmetric
and an antisymmetric skew-polynomial.
25
It suffices to show all An -invariants that are not Sn -invariant can be constructed
in this manner. Let f ∈ RAn and not in RSn . Therefore, some odd permutation σ
has the property σf 6= f . Consider g = f + σf and h = f − σf . It is clear that
1
2
(g + h) = f , so it remains to show that g is a Sn -invariant and h is antisymmetric.
Lemma 4.1.5. The An -invariant g is a Sn -invariant.
Proof. Let τ ∈ Sn − An , and consider
τ (g) = τ (f ) + τ (σ(f ))
= σ(f ) + f
= g.
Let τ ∈ An , and consider
τ (g) = τ (f ) + τ (σ(f ))
= σ(f ) + f
= g.
Thus g is a Sn -invariant.
Lemma 4.1.6. The An -invariant h is antisymmetric.
Proof. Let τ be some odd permutation, and consider
τ (h) = τ (f ) − τ (σ(f ))
= τ (f ) − f
= −(f − σ(f ))
= −h.
Hence, h is antisymmetric.
Therefore every An -invariant is a sum of a Sn -invariant and an antisymmetric invariant.
26
Now we want to show the decomposition is unique.
Proof. Let A1 and A2 be antisymmetric and S1 and S2 be symmetric. Assume for
contradiction that f = A1 + S1 and f = A2 + S2 . So A1 + S1 = A2 + S2 , thus
A1 − A2 = S2 − S1 . Therefore A1 − A2 is a symmetric polynomial. Let σ be any odd
permutation and consider σ acting on (A1 − A2 ).
σ(A1 − A2 ) = σ(A1 ) − σ(A2 )
= −A1 + A2
= −(A1 − A2 )
= A1 − A2 .
Thus, A1 = A2 , and hence S1 = S2 . Therefore, f decomposes uniquely into the sum
of an antisymmetric invariant and a symmetric invariant.
Lemma 4.1.7. Linear combinations of antisymmetric invariants are antisymmetric
Proof. Let f and g be antisymmetric, c, d ∈ k and let τ be any odd permutation
τ (cf + dg) = τ (cf ) + τ (dg)
= cτ (f ) + dτ (g)
= −cf − dg
= −(cf + dg).
If f and f + g are antisymmetric then g is antisymmetric too. Also if f = gh and
f and g are antisymmetric then h is symmetric.
n
Lemma 4.1.8. The invariants OAn (Πn−1
i=1 xi ) and OAn (Πi=1 xi ) are antisymmetric.
27
Proof. First we show OAn (Πni=1 xi ) is antisymmetric. To show this let’s show that any
transposition, (i, j), acting on OAn (Πni=1 xi ) satisfies (i, j)OAn (Πni=1 xi ) = −OAn (Πni=1 xi ).
The invariant OAn (Πni=1 xi ) = Πni=1 xi . Without loss of generality let i > j and let (i, j)
be any transposition and compute:
(i, j)OAn (Πni=1 (xi ) = (i, j)(x1 x2 · · · xi · · · xj · · · xn )
= x1 x2 · · · xj · · · xi · · · xn
= (−1)2(j−i−1)+1 (x1 x2 · · · xi · · · xj · · · xn )
= −(x1 x2 · · · xi · · · xj · · · xn )
= −OAn (Πni=1 (xi )
The fact that any odd permutation is the product of an odd number of transpositions and that any transposition takes OAn (Πni=1 (xi ) to −OAn (Πni=1 (xi ), we have that
OAn (Πni=1 (xi ) is antisymmetric.
Recall that if f is symmetric, g is antisymmetric, and f = gh, then h is antisymmetn
2
2 2
ric. We note that ±OAn (Πni=1 xi )OAn (Πn−1
i=1 xi ) = OSn (x1 x2 · · · xn−1 xn ), OAn (Πi=1 (xi )
n−1
is antisymmetric, and OSn (x21 x22 · · · x2n−1 xn ) is symmetric. Therefore, OAn (Πi=1
(xi )
is antisymmetric.
4.2
RAn Generators
We now proceed to the discussion of producing algebra generators of RAn .
Definition 8. An exponent sequence I = (ij ) is a pseudo-partition if ij ≥ ij+1 for
1 ≤ j ≤ n − 2 and in−2 > in > in−1 . We denote the pseudo-partition as λ(I).
Example 4.2.1. Consider the ring k−1 [x1 , x2 , x3 ]A3 , and the orbit sums OA3 (x1 ) and
OA3 (x31 x2 x23 ).
The orbit sum OA3 (x1 ) = x1 + x2 + x3 . The monomial x1 occurs in the orbit sum;
28
therefore, we can define the orbit sum with the partition (1, 0, 0).
Now the orbit sum OA3 (x31 x2 x23 ) = x31 x2 x23 +x21 +x32 x3 −x1 x22 x33 . The monomial x31 x22 x3
does not occur in the orbit sum so we cannot define this orbit sum with the partition
(3, 2, 1), yet the monomial x31 x2 x23 occurs in the obit sum. Hence we can define this
orbit with the pseudo-parition (3, 1, 2).
Lemma 4.2.2. Every orbit sum under the alternating group is represented by a monomial X I where I is either a partition or a pseudo-partition.
Proof. For any exponent sequence I there exists some permutation σ such that
σ(I) = λ(I), a partition. If σ ∈ An then X λ(I) is a summand in OAn (X I ) and
OAn (X I ) = OAn (X λ(I) ). Otherwise σ ∈ Sn − An and therefore (n − 1, n)σ ∈
An . The permutation (n − 1, n)σ induces a pseudo-partition where acting on I,
(n − 1, n)σ(I) = (λ1 (I), λ2 (I), . . . , λn−2 (I), λn (I), λn−1 (I)) and λn (I) < λn−1 (I)).
Similarly, OAn (X I ) = OAn (X λ(I) ). So for any orbit sum under the alternating group,
OAn (X I ), we can assume that the exponent sequence I of the representative monomial
X I is a partition or a pseudo-partition.
The symmetric group Sn = An ∪ σAn for any odd permutation σ. Therefore
OrbSn (X I ) = OrbAn (X I ) ∪ σOrbAn (X I ). This implies that OSn (X I ) = OAn (X I ) or
OSn (X I ) = OAn (X I ) + σOAn (X I ).
There are two trivial cases when determining the algebra generators for RAn . The
first trivial case is the subring RA1 = RS1 = R, and the second is RA2 = R. Both of
these subrings are equal to R, and so we already know the generators for the subring.
Lemma 4.2.3. The invariant OAn (Πni=1 xi ) = Πni=1 xi can be constructed by OAn (x1 )
and OAn (Πn−1
i=1 xi ) for n ≥ 3.
Proof. First consider when n = 3. Compute OA3 (Π2i=1 xi )OA3 (x1 ) = 3OA3 (Π3i=1 xi ) +
OA3 (x21 x2 ) − OAn (x21 x3 ), and
29
OA3 (x1 )OA3 (Π2i=1 xi ) = 3OA3 (Π3i=1 xi ) − OA3 (x21 x2 ) + OAn (x21 x3 ). So,
1
6
(OA3 (Π2i=1 xi )OA3 (x1 ) + OA3 (x1 )OA3 (Π2i=1 xi )) = Π3i=1 xi . Now let n ≥ 4, and no-
tice that for any orbit sum of a monomial X I where I is sequence of n − 1 ones
and 1 zero, the orbit sum is defined by the partition λ(I) = (1, 1, . . . , 1, 0) and
not the pseudo-partition λ(I) = (1, 1, . . . , 1, 0, 1). This is because n ≥ 4 so n −
1 > 2, thus, the even permutation (1,2)(n-1,n) acting on X λ(I) = ±X λ(I) . Therefore, the only partitions and pseudo-partitions we need to count from the product
OAn (Πn−1
i=1 xi )OAn (x1 ) are (2, 1, . . . , 1, 0) and (1, 1, . . . , 1). The only way to get the
n−1
partition (2, 1, . . . , 1, 0) is by (Πi=1
xi )(x1 ). The sign of this monomial is (−1)n−2 . The
n
other sequence (1, 1, . . . , 1) occurs n many times, namely ((−1)n−j Πj−1
i=1 xi Πi=j+1 xi )(xj )
for all 1 ≤ j ≤ n. The sign of any of these monomials is (−1)n−j (−1)n−j = 1.
n−2
Hence, OAn (Πn−1
OAn (x21 x2 · · · xn−1 )+nOAn (Πni=1 xi ). Similarly,
i=1 xi )OAn (x1 ) = (−1)
n−1
2
nOAn (Πni=1 xi ). When n ≥ 4 and
OAn (xi )OAn (Πn−1
i=1 xi ) = OAn (x1 x2 · · · xn−1 ) + (−1)
n−1
1
n
OAn (Πn−1
n is even we get 2n
i=1 xi )OAn (x1 ) − OAn (xi )OAn (Πi=1 xi ) = Πi=1 xi . When
n−1
1
n
n is odd we get 2n
OAn (Πn−1
i=1 xi )OAn (x1 ) + OAn (xi )OAn (Πi=1 xi ) = Πi=1 xi . Thus,
n−1
xi ).
OAn (Πni=1 xi ) = Πni=1 xi can be constructed by OAn (x1 ) and OAn (Πi=1
Corollary 4.2.4. The Sn -invariant OSn (x21 x22 · · · x2n−1 xn ) can be generated by
OAn (Πn−1
i=1 xi ) and OAn (x1 ).
n−1
xi ). From Lemma
Proof. The invariant OSn (x21 x22 · · · x2n−1 xn ) = ±OAn (Πni=1 xi )OAn (Πi=1
4.2.3, OAn (Πni=1 xi ) = Πni=1 xi can be constructed by OAn (x1 ) and OAn (Πn−1
i=1 xi ). Therefore, the invariant OSn (x21 x22 · · · x2n−1 xn ) can be constructed by OAn (Πn−1
i=1 xi ) and
OAn (x1 ).
Lemma 4.2.5. All antisymmetric orbit sums OAn (X I ) are represented by a monomial
X I where I is a partition.
Proof. Let OAn (X I ) be an orbit sum that is an antisymmetric invariant. Then from
30
Lemma 4.2.1 I is either a partition or a pseudo-partition. If I is a partition then
we are done. Otherwise I is a pseudo-partition. Then consider the odd permutation
(n, n − 1) acting on OAn (X I ), (n, n − 1)OAn (X I ) = −OAn (X λ(I) ). Thus −OAn (X I )
is represented by a monomial whose exponent sequence is a partition.
Lemma 4.2.6. Every antisymmetric orbit sum OAn (X I ) has a pair of repeated odd
entries in I.
Proof. Let OAn (X I ) be an orbit sum that is an antisymmetric invariant. Therefore
any odd permutation, σ, has the property σOAn (X I ) = −OAn (X I ). Thus no odd
permutations stabilize OAn (X I ). By Lemma 4.1.1. OAn (X I ) 6= OSn (X I ). Thus
OSn (X I ) = OAn (X I ) + σOAn (X I ). Therefore:
OSn (X I ) = OAn (X I ) + σOAn (X I )
= OAn (X I ) − OAn (X I )
= 0.
Thus the orbit sum under Sn is trivial. Lemma 2.2.2 implies that the exponent
sequence of X I has a pair of repeated odd entries.
Theorem 4.2.7. The ring RAn is generated as an algebra by the first n − 1 skewsymmetric polynomials and OAn (Πn−1
i=1 xi ).
Proof. Since every invariant under An can be written as a linear combination of antisymmetric invariants and Sn -invariants, it suffices to show that any antisymmetric
invariants f can be constructed in this manner.
Let f be antisymmetric. Therefore, by Lemma 4.2.3, f has a leading monomial
X I where I = (i1 , i2 , . . . , in ) and I is a partition. If in 6= 0 then let I e = (i1 −
e
1, i2 − 1, . . . , in − 1). The fact that OAn (X I ) 6= 0 implies OAn (X I ) 6= 0. Therefore,
e
OAn (X I ) = ± (Πni=1 xi ) OAn (X I ). So we may assume that in = 0.
31
Consider I ∗ = (i1 − 1, i2 − 1, . . . , in−1 − 1, 0). Since OAn (X I ) is antisymmetric then
Lemma 4.2.6, implies that I has a pair of repeated odd entries. Thus I ∗ has a
∗
pair of repeated even entries. By Lemma 4.1.1. OAn (X I ) is a Sn -invariant. Since
OAn (X I ) is not a Sn -invariant, by Lemma 4.1.1, the exponent sequence I has no repeated even entries. Thus I ∗ has no repeated odd entries. Therefore by Lemma 4.1.2.
∗
∗
n−1
(xi ))OAn (X I ). Since OAn (Πn−1
OAn (X I ) 6= 0. Consider OAn (Πi=1
i=1 (xi )) is antisym∗
∗
I
metric and OAn (X I ) is a Sn -invariant, we can conclude that OAn (Πn−1
i=1 (xi ))OAn (X )
∗
I
is antisymmetric. Therefore, OAn (Πn−1
i=1 (xi ))OAn (X ) has a leading monomial under
the lexicographic order, which is X I , and all other monomials are smaller lexico∗
I
graphically. Thus, all summands occurring in OAn (X I ) ± OAn (Πn−1
i=1 (xi ))OAn (X )
are smaller lexicographically and are antisymmetric. Thus there are only a finite
number of degree sequences less than I, so this process terminates. Hence, every
antisymmetric orbit sum can be written in terms of the first n − 1 elementary skewsymmetric invariants and OAn (Πn−1
i=1 xi ).
32
Chapter 5:
Examples With Göbel’s Bound with 4
indeterminates
The intent of this chapter is to compute explicit examples of some invariant subrings. We explore subrings of R = k−1 [x1 , x2 , x3 , x4 ], the skew-polynomial ring with
four indeterminates, since all the other invariant subrings with three or less indeterminates have been computed by previous theorems in this thesis. Recall that Göbel’s
bound in the commutative case showed that for four indeterminates, generators of
RG have degree less than or equal to 42 =6. Also, Göbel’s bound was sharp in the
commutative case since k[x1 , x2 , x3 , x4 ]A4 needs a generator of degree 6, namely the
Vandermonde Determinant. In this chapter we explore Göbel’s bound to determine
how sharp the bound is in the non-commutative case.
5.1
Alternating Z2 Subgroup
Let Z2 be a subgroup of S4 defined by:

1 0 0



0 1 0
Z2 = 

0 0 1



0 0 0
 
0
0 1


0   1 0
,
0   0 0
1
0 0
0
0
0
1

0 


0 
 .
1 


0
This Z2 subgroup is contained in A4 and is generated by g = (1, 2)(3, 4).
Lemma 5.1.1. An orbit sum OZ2 (X I ) = 0 if and only if I = (i1 , i2 , i3 , i4 ) where
i1 = i2 , i3 = i4 and either i1 is even and i3 is odd or i1 is odd and i3 is even.
33
Proof. ⇒) Let OZ2 (X I ) = 0, so g(X I ) = −X I . Therefore, g(I) = I
g(I) = g(i1 , i2 , i3 , i4 )
= (i2 , i1 , i4 , i3 )
= (i1 , i2 , i3 , i4 )
Therefore, i1 = i2 and i3 = i4 . So g(xi11 xi21 xi33 xi43 ) = (−1)i1 (−1)i3 (xi11 xi21 xi33 xi43 ). Hence,
either i1 or i3 must be odd, but not both.
⇐). Consider OZ2 (X I ) and let I = (i1 , i1 , i3 , i3 ) and i1 is even and i3 is odd or vice
versa. Then OZ2 (X I ) = X I +g(X I ) = X I −X I = 0. Thus the orbit sum is trivial.
To determine a set of generators for RZ2 we use Göbel’s bound along with the
generators from RA4 to bound the degree of potential generators. To further bound
the degree of potential generators we take OZ2 (x21 x22 ), OZ2 (x23 x4 ), and OZ2 (x1 x2 x3 ) to
bound generators. Let OZ2 (X I ) be a potential generator of RZ2 .
1. Göbel’s bound bounds the degree of a generator of OZ2 (X I ) to 16 where λ(I) =
(λ1 (I), λ2 (I), λ3 (I), λ4 (I)), λi − λi+1 ≤ 2 and λ4 (I) = 1 or 0.
2. When λ4 (I) = 1 we consider I ∗ = (i1 − 1, i2 − 1, i3 − 1, i4 − 1), and notice
∗
±OZ2 (X I )OA4 (x1 x2 x3 x4 ) = OZ2 (X I ). Therefore, we may assume that
λ4 (I) = 0.
3. When λ3 (I) = 2, the exponents of x1 and x2 are both at least 2, or the exponent
of x3 and x4 are at least two. Therefore, OZ2 (xi11 −2 xi22 −2 xi33 xi44 )OZ2 (x21 x22 ) =
OZ2 (X I ), or OZ2 (xi11 xi22 xi33 −2 xi44 −2 )OZ2 (x23 x24 ) = OZ2 (X I ). Therefore, we may
assume that λ3 (I) ≤ 1.
At this point we have reduced possible generators, OZ2 (X I ), to have the associated
partition of λ(I) = (λ1 (I), λ2 (I), 0, 0) or (λ1 (I), λ2 (I), 1, 0) and be special. We further
34
bound the case when λ3 (I) = 1 and λ2 (I) ≥ 2. We break this case into two subcases:
1) the exponents of x1 and x2 are both at least 2 or the exponent of x3 and x4 are at
least 2, and 2) otherwise.
1. When the exponents of x1 and x2 are both at least 2 or the exponent of x3 and
x4 are at least 2, then either OZ2 (xi11 −2 xi22 −2 xi33 xi44 )OZ2 (x21 x22 ) = OZ2 (X I ),
or OZ2 (xi11 xi22 xi33 −2 xi44 −2 )OZ2 (x23 x24 ) = OZ2 (X I ).
2. Let at least one of the exponents of x1 and x2 be less than 2 and at least one of
λ (I) λ (I)
2
1
xσ(3) x0σ(4) . Conxσ(2)
the exponents of x3 and x4 be less than 2. Let X I = ±xσ(1)
∗
λ (I)−1 λ2 (I)−1
xσ(2) .
1
sider ±X I = xσ(1)
So both non-zero exponents cannot be the expo-
nents of x1 and x2 nor can they both be the exponents of x3 and x4 . Therefore,
∗
∗
by Lemma 5.1.1, OZ2 (X I ) 6= 0. Consider ±OZ2 (X I )OZ2 (xσ(1) xσ(2) xσ(3) ) =
OZ2 (X I )±OZ2 (xe11 xe22 xe33 xe44 ), where every ei > 0. So the orbit sums OZ2 (xe11 xe22 xe33 xe44 )
is reducible and thus, we can construct OZ2 (xe11 xe22 xe33 xe44 ). Therefore, we can
assume λ3 (I) ≤ 1.
So possible partitions for exponent sequences of potential generators are from the
set S = {(1, 0, 0, 0), (2, 0, 0, 0), (1, 1, 0, 0), (2, 1, 0, 0), (2, 2, 0, 0), (3, 1, 0, 0), (3, 2, 0, 0),
(4, 2, 0, 0), (1, 1, 1, 0), (2, 1, 1, 0), and (3, 1, 1, 0)}.
Theorem 5.1.2. The set of invariants G = {OZ2 (x21 x22 ), OZ2 (x23 x4 ), OZ2 (x1 x2 x3 ), OA4 (x1 ),
OA4 (x21 x2 ), OA4 (x21 x22 x3 ), OZ2 (x1 ), OZ2 (x1 x3 ), OZ2 (x21 x2 ) and OA4 (x1 x2 x3 )} generate RZ2 .
Proof. The remaining invariants, OZ2 (X I ), where the associated partition λ(I) is an
element of S are shown explicitly in the appendix, and thus, the theorem is proved.
35
5.2
Alternating Klein IV Subgroup
Let V be a Klein-4 subgroup of S4 generated by matrices with representation:

1 0 0



0 1 0
V= 
 0 0 1



0 0 0
 
0
0 1


0   1 0
,
0   0 0
1
0 0
0
0
0
1
 
0
0 0


0   0 0
,
1   1 0
0
0 1
1
0
0
0
 
0
0 0


1   0 0
,
0   0 1
0
1 0
0
1
0
0

1 


0 

0 


0
We will denote V as the set {(1), (12)(34), (13)(24), (14)(23)}. This particular V
is a subgroup contained in A4 , so we can use the generators for RA4 to help bound
the degree of generators needed for RV .
Lemma 5.2.1. The orbit sum OV (X I ) = OA4 (X I ) if and only if there exists some
three cycle c such that c(X I ) = X I .
Before we can prove this claim we must prove a lemma first.
Lemma 5.2.2. A4 can be generated by the set {(12)(34), (13)(24), (14)(23), c}for any
three cycle c in A4 .
Proof. Clearly the subgroup generated by the set {(12)(34), (13)(24), (14)(23), c} contains the four elements shown, the inverse of c, and the identity. This accounts for six
elements. The group A4 has no subgroups of order 6 so {(12)(34).(13)(24), (14)(23), c}
must generate A4 .
Lemma 5.2.1. ⇒) Let OV (X I ) = OA4 (X I ), and assume for contradiction that every
three cycle c has the property c(X I ) 6= X I . Since OV (X I ) = OA4 (X I ), we have that
OrbV (X I ) = OrbA4 (X I ). Thus, the StabA4 (X I ) ⊆ {(12)(34), (13)(24), (14)(23)} = V.
This implies StabA4 (X I ) = StabV (X I ). We know |V| = |StabV ||OrbV (X I )|, and
|A4 | = |StabA4 (X I )||OrbA4 (X I )|. Since OrbV (X I ) = OrbA4 (X I ) and StabA4 (X I ) =
StabV (X I ), we can substitute and get |V| = |StabA4 (X I )||OrbA4 (X I )| = |A4 |. Hence,
36
|V| = |A4 | implies 4=12, but 4 6= 12. Therefore some three cycle, c, must have the
property c(X I ) = X I .
⇐) Let c be a three cycle such that c(X I ) = X I . We also know A4 is gnerated
by the set {(12)(34), (13)(24), (14)(23), c}. So OrbA4 (X I ) ⊆ V ∪ Vc ∪ Vc2 . Thus
OrbA4 (X I ) ⊆ OrbV (X I ). V ≤ A4 therefore we have that OrbV (X I ) ⊆ OrbA4 (X I ).
Hence, OrbV (X I ) = OrbA4 (X I ), and therefore OV (X I ) = OA4 (X I ).
Theorem 5.2.3. The orbit sum OV (X I ) = 0 if and only if OA4 (X I ) = 0.
Proof. ⇒) Let OV (X I ) = 0. Then there exists g ∈ V such that g(X I ) = −X I .
Since V ≤ A4 implies that g ∈ A4 , −X I is a term occurring in OA4 (X I ). Thus
OA4 (X I ) = 0.
⇐) Let OA4 (X I ) = 0. This implies that the exponent sequence I has a repeated
even and repeated odd entry. So any permutation σ in A4 that takes X I to −X I
must interchange the even exponents and interchange the odd exponents. Therefore,
σ must be the product of two disjoint transpostitions within A4 . So σ must also be
in V. Therefore −X I must occur in OV (X I ). Thus OV (X I ) = 0.
We want to determine the generators by bounding the degree of monomials. We
can look explicitly at the associated partition of the exponent sequence on a monomial. Göbel’s Bound bounds the degree of a monomial, X I , to 16 with an associated
partition of the form λ(I) = (λ1 , λ2 , λ3 λ4 ), where λi − λi+1 ≤ 2 and λ4 is either 1 or
0.
When λ4 = 1, then we can factor out OV (x1 x2 x3 x4 ) = x1 x2 x3 x4 from OV (X I ).
∗
This results in OV (X I ) = ±OV (x1 x2 x3 x4 )OV (X I ), where λ(I ∗ ) = (λ1 (I) − 1, λ2 (I) −
1, λ3 (I)−1, 0). So we can assume that for any generator OV (X I ) of RV has λ4 (I) = 0,
which bounds the degree to 12.
37
When λ3 = 2 we consider λ(I ∗ ) = (λ1 (I) − 1, λ2 (I) − 1, 1, 0). Then, consider
∗
∗
∗
∗
OV (X I )OV (x1 x2 x3 ) = OV (X I )(x1 x2 x3 ) − OV (X I )(x1 x2 x4 ) + OV (X I )(x1 x3 x4 ) −
∗
OV (X I )(x2 x3 x4 ). The fact that I ∗ has exactly three non-zero entries implies that
∗
exactly four monomials from OV (X I )OV (x1 x2 x3 ) will have exactly three non-zero
entries in their exponent sequence. These four monomials will form OV (X I ). Every
∗
other monomial in OV (X I )OV (x1 x2 x3 ) will have an exponent sequence K where K
has 4 non-zero entries. Thus, we can reduce them by the previous step. Now we can
assume that for any generator OV (X I ) of RV has λ3 (I) ≤ 1, which bounds the degree
to 9.
So possible partitions for exponent sequences of potential generators are from the
set S = {(1, 0, 0, 0), (2, 0, 0, 0), (1, 1, 0, 0), (2, 1, 0, 0), (2, 2, 0, 0), (3, 1, 0, 0), (3, 2, 0, 0),
(4, 2, 0, 0), (1, 1, 1, 0), (2, 1, 1, 0), (3, 1, 1, 0), (2, 2, 1, 0), (3, 2, 1, 0), (3, 3, 1, 0), (4, 2, 1, 0),
(4, 3, 1, 0), and (5, 3, 1, 0)}.
Theorem 5.2.4. The ring RV is generated by G = {OS4 (x1 ), OS4 (x21 x2 ), OS4 (x21 x22 x3 ),
OA4 (x1 x2 x3 ), OV (x21 x2 ), OV (x21 x3 ), OV (x31 x2 ), OV (x31 x3 )}.
Proof. The remaining invariants, OV (X I ), where the associated partition λ(I) is an
element of S are shown explicitly in the appendix, and thus, the theorem is proved.
5.3
Non-Alternating Klein IV Subgroup
Let V be a Klein-4 subgroup of S4 generated by matrices with representation:

1 0 0



0 1 0
V= 

0 0 1



0 0 0
 
0 1
0
 1 0
0 
,
0   0 0
0 0
1
0
0
1
0
 
0
1 0
 0 1
0 
,
0   0 0
1
0 0
38
0
0
0
1
 
0
0 1
 1 0
0 
,
1   0 0
0
0 0
0
0
0
1

0 


0 

1 


0
We will denote V as V = {(1), (1, 2), (3, 4), (1, 2)(3, 4)} and note that V is generated by (1, 2) and (3, 4).
Lemma 5.3.1. R(1,2) ∩ R(3,4) = RV .
Proof. Let f ∈ R(1,2) ∩ R(3,4) . Thus (1, 2)f = f and (3, 4)f = f . Hence f ∈ RV .
Conversely, let f ∈ RV , so (1, 2)f = (3, 4)f = f . So f ∈ R(1,2) ∩ R(3,4) . Therefore,
R(1,2) ∩ R(3,4) = RV .
Therefore, a natural choice for generators of RV would be the generators for R(1,2)
that are also invariant under (3, 4), and the generators for R(3,4) that are also invariant
under (1, 2). These generators would be (x1 + x2 ) = g1 , (x3 + x4 ) = g2 , (x31 + x32 ) = g3 ,
and (x33 + x34 ) = g4 . Let G = {g1 , g2 , g3 , g4 }.
Lemma 5.3.2. Given an orbit sum OV (X I ), if i3 = i4 and i3 is odd, then OV (X I ) =
0.
2
Proof. Consider (3, 4)X I = (3, 4)xi11 xi22 xi33 xi43 = (−1)i3 )xi11 xi22 xi33 xi43 = −X I .
So, OV (X I ) = X I + (1, 2)X I + (3, 4)X I + (1, 2)(3, 4)X I = X I + (1, 2)X I − X I −
(1, 2)X I = 0.
Hence, we can assume that for any arbitrary polynomial f ∈ RV has no monomial
X I where i3 = i4 and i3 is odd.
Theorem 5.3.3. The ring RV is generated as an algebra by G.
Proof. ⊇) Clearly, every polynomial generated from G is invariant under V since the
generators are invariant under both (1, 2) and (3, 4).
⊆) Let f ∈ RV . We can consider rewriting R as a polynomial ring with indeterminates x3 and x4 , with coefficients as polynomials in terms of x1 and x2 , denoted
k−1 [x1 , x2 ](x3 , x4 ). Since f ∈ k−1 [x1 , x2 ](x3 , x4 ), f can be written as:
39
f=
X
pi,j (x1 , x2 )xi3 xj4 ,
i,j
where pi,j (x1 , x2 ) is some polynomial in terms of x1 and x2 . Since f ∈ RV , (3, 4)f = f .
X
(3, 4)f = (3, 4)
pi,j (x1 , x2 )xi3 xj4
i,j
=
X
(3, 4)pi,j (x1 , x2 )(3, 4)xi3 xj4
i,j
=
X
(−1)ij pi,j (x1 , x2 )xj3 xi4
i,j
= f.
This shows pi,j (x1 , x2 ) = (−1)ij pj,i (x1 , x2 ). Now consider (1, 2) acting on f :
X
(1, 2)f = (1, 2)
pi,j (x1 , x2 )xi3 xj4
i,j
=
X
(1, 2)pi,j (x1 , x2 )(1, 2)xi3 xj4
i,j
=
X
(1, 2)pi,j (x1 , x2 )xi3 xj4
i,j
= f.
Thus pi,j (x1 , x2 ) = (1, 2)pj,i (x1 , x2 ), hence pi,j (x1 , x2 ) ∈ R(1,2) , and pi,j (x1 , x2 ) is gen-
40
erated by g1 and g3 . Now we rewrite f as:
X
X
f=
pi,i (x1 , x2 )xi3 xi4 .
pi,j (x1 , x2 )xi3 xj4 + pj,i (x1 , x2 )xj3 xi4 +
i,i
i>j
=
X
X
pi,j (x1 , x2 )xi3 xj4 + (−1)ij pi,j (x1 , x2 )xj3 xi4 +
pi,i (x1 , x2 )xi3 xi4 .
i>j
=
X
i,i
pi,j (x1 , x2 ) xi3 xj4 + (−1)ij xj3 xi4
i>j
+
X
pi,i (x1 , x2 )xi3 xi4 .
i,i
= f.
Let (3, 4) act on a term (xi3 xj4 + (−1)ij xj3 xi4 ).
(3, 4)(xi3 xj4 + (−1)ij xj3 xi4 ) = ((−1)ij xj3 xi4 + (−1)2ij xi3 xj4 )
= (xi3 xj4 + (−1)ij xj3 xi4 ).
Thus,(xi3 xj4 + (−1)ij xj3 xi4 ) is an element of R(3,4) , and (xi3 xj4 + (−1)ij xj3 xi4 ) is generated
2
by g2 and g4 . Now consider (3, 4)(xi3 xi4 ) = (−1)i xi3 xi4 = xi3 xi4 . Therefore, (xi3 xi4 ) is in
R(3,4) , and is generated by g2 and g4 . Hence f is generated by G.
41
Bibliography
[1] H. Derksen, G. Kemper. Computational Invariant Theory, volume 130 Encyclopaedia of Mathematical Sciences. Springer, Berlin, Gemany, 2002. Invariant
Theory and Algebraic Transformation Groups.
[2] Mara D. Neusel. Invariant Theory, volume 36 of Student Mathematical Library.
American Mathematical Society, Providence, Rhode Island, 2006.
[3] Bernd Sturmfels.
Algorithms in Invariant Theory, 2nd edition.
Verlag/Wien, Germany, 2008.
42
Springer-
Appendix A:
5.1
The remaining invariants that need to be explicitly generated by G for Theorem
5.1.2 are listed here.
OZ2 (x1 ) ∈ G
OZ2 (x3 ) = OA4 (x1 ) − OZ2 (x1 ).
OZ2 (x21 ) = OZ2 (x1 )2
OZ2 (x23 ) = OZ2 (x3 )2
OZ2 (x1 x3 ) ∈ G
OZ2 (x1 x4 ) = OZ2 (x1 )OZ2 (x3 ) − OZ2 (x1 x3 )
OZ2 (x1 x2 x3 ) =
OZ2 (x21 x3 ) =
−1
2
−1
2
(OZ2 (x1 x3 )OZ2 (x1 ) + OZ2 (x1 )OZ2 (x1 x3 ))
(OZ2 (x1 x3 )OZ2 (x1 ) − OZ2 (x1 )OZ2 (x1 x3 ))
OZ2 (x1 x3 x4 ) = 21 (OZ2 (x1 x3 )OZ2 (x3 ) + OZ2 (x3 )OZ2 (x1 x3 ))
OZ2 (x1 x2 x3 ) =
−1
2
(OZ2 (x1 x3 )OZ2 (x1 ) + OZ2 (x1 )OZ2 (x1 x3 ))
OZ2 (x1 x23 ) = 21 (OZ2 (x1 x3 )OZ2 (x3 ) − OZ2 (x3 )OZ2 (x1 x3 ))
OZ2 (x21 x2 ) ∈ G
OZ2 (x23 x4 ) = OZ2 (x3 )3 − OA4 (x31 ) + OZ2 (x31 ) − O( x21 x2 )
OZ2 (x21 x4 ) = (OZ2 (x1 )2 OZ2 (x3 )) + 12 (OZ2 (x1 x3 )OZ2 (x1 ) − OZ2 (x1 )OZ2 (x1 x3 ))
OZ2 (x1 x24 ) = (OZ2 (x1 )OZ2 (x3 )2 ) − 21 (OZ2 (x1 x3 )OZ2 (x3 ) − OZ2 (x3 )OZ2 (x1 x3 ))
OZ2 (x21 x2 x3 ) = 21 (OZ2 (x1 x2 x3 )OZ2 (x1 ) + OZ2 (x1 )OZ2 (x1 x2 x3 ))
OZ2 (x1 x22 x3 ) =
−1
(OZ2 (x1 x2 x3 )OZ2 (x1 )
2
− OZ2 (x1 )OZ2 (x1 x2 x3 ))
OZ2 (x1 x3 x24 ) = 21 (OZ2 (x1 x3 x4 )OZ2 (x3 ) − OZ2 ((x3 x24 )OZ2 (x1 ))
OZ2 (x21 x2 x4 ) = 21 (OZ2 (x1 x2 x4 )OZ2 (x1 ) + OZ2 (x21 x2 )OZ2 (x3 ))
OZ2 (x21 x3 x4 ) = OZ2 (x1 x3 x4 )OZ2 (x1 ) − OZ2 (x1 x2 x3 x4 )
OZ2 (x1 x2 x23 ) = OZ2 (x1 x2 x3 )OZ2 (x3 ) − OZ2 (x1 x2 x3 x4 )
OZ2 (x1 x23 x4 ) = −OZ2 (x1 x3 x4 )OZ2 (x3 ) + OZ2 (x1 x2 x24 )
OZ2 (x21 x23 ) = −OZ2 (x1 x3 )2
43
OZ2 (x21 x24 ) = 12 (OZ2 (x1 )4 − OA4 (x41 ))
OZ2 (x23 x24 ) = 12 (OZ2 (x3 )4 − OA4 (x41 )
OZ2 (x21 x24 ) = −OZ2 (x1 x4 )2
OZ2 (x31 x2 ) = OZ2 (x1 )OZ2 ((x21 x2 ) − OZ2 (x21 x22 )
OZ2 (x33 x4 ) = OZ2 (x3 )OZ2 (x23 x4 ) − OZ2 (x23 x24 )
OZ2 (x31 x3 ) = OZ2 (x21 )OZ2 (x1 x3 ) − OZ2 (x1 x22 x3 )
OZ2 (x31 x4 ) = OZ2 (x21 )OZ2 (x1 x4 ) − OZ2 (x1 x22 x4 )
OZ2 (x1 x33 ) = OZ2 (x23 )OZ2 (x1 x3 ) − OZ2 ((x1 x3 x24 )
OZ2 (x1 x34 = OZ2 (x24 )OZ2 (x1 x4 ) − OZ2 (x1 x23 x4 )
OZ2 (x31 x2 x3 ) = −OZ2 (x21 x2 )OZ2 (x1 x3 ) + OZ2 (x21 x22 )OZ2 (x3 )
OZ2 (x31 x2 x4 ) = −OZ2 (x21 x2 )OZ2 (x1 x4 ) + OZ2 (x21 x22 )OZ2 (x4 )
OZ2 (x1 x33 x4 ) = OZ2 (x23 x4 )OZ2 (x1 x3 ) + OZ2 (x23 x24 )OZ2 (x2 )
OZ2 (x1 x3 x34 = −OZ2 (x3 3x24 )OZ2 (x1 x4 ) − OZ2 (x23 x24 )OZ2 (x1 )
OZ2 (x31 x3 x4 ) = −OZ2 (x21 x3 )OZ2 (x1 x4 ) − OZ2 (x21 x2 x23 )
OZ2 (x1 x2 x23 ) = −OZ2 (x2 x24 )OZ2 (x1 x4 ) − OZ2 (x22 x3 x24 )
OZ2 (x31 x22 ) = OZ2 (x21 x22 )OZ2 (x1 )
OZ2 (x33 x24 ) = OZ2 (x23 x24 )OZ2 (x3 )
OZ2 (x31 x23 ) = OZ2 (x21 )OZ2 (x1 x23 ) − OZ2 (x1 x22 x23 )
OZ2 (x31 x24 ) = OZ2 (x21 )OZ2 (x1 x24 ) − OZ2 (x1 x22 x24 )
OZ2 (x21 x33 ) = OZ2 (x21 x3 )OZ2 (x23 ) − OZ2 (x21 x3 x24 )
OZ2 (x21 x34 ) = OZ2 (x21 x4 )OZ2 (x24 ) − OZ2 (x21 x22 x4 )
OZ2 (x41 x22 ) = OZ2 (x21 x22 )OZ2 (x21 )
OZ2 (x43 x24 ) = OZ2 (x23 x24 )OZ2 (x23 )
OZ2 (x41 x23 ) = OZ2 (x21 x3 )2 − OZ2 (x21 x22 x3 x4 )
OZ2 (x41 x24 ) = OZ2 (x21 x4 )2 + OZ2 (x21 x22 x3 x4 )
OZ2 (x21 x24 ) = OZ2 (x1 x23 )2 − OZ2 (x1 x2 x23 x24 )
44
OZ2 (x21 x44 ) = OZ2 (x1 x24 )2 − OZ2 (x1 x2 x23 x24 )
45
Appendix B:
5.2
The remaining invariants that need to be explicitly generated by G for Theorem
5.2.4 are listed here.
OV (x1 ) ∈ G
OV (x21 ) = OV (x1 )2
OV (x31 ) ∈ RA4
OV (x21 x2 ) ∈ G
OV (x21 x3 ) ∈ G
OV (x21 x4 ) = OA4 (x21 x2 ) − OV (x21 x2 ) − OV (x21 x3 )
OV (x1 x2 x3 ) ∈ G
OV (x41 ) = 12 (OV (x31 )OV (x1 ) + OV (x1 )OV (x31 ))
= OV (x1 )4 − OV (x21 x2 )OV (x1 ) − OV (x21 x3 )OV (x1 ) − OV (x21 x4 )OV (x1 )
OV (x31 x2 ) ∈ G
OV (x31 x3 ) ∈ G
OV (x31 x4 ) = OA4 (x31 x2 ) − OV (x31 x2 ) − OV (x31 x3 )
OV (x21 x22 ) = 41 (OV (x21 x2 )OV (x1 ) + OV (x1 )OV (x31 x2 ))
OV (x21 x23 ) = 41 (OV (x21 x3 )OV (x1 ) + OV (x1 )OV (x31 x3 ))
OV (x21 x24 ) = 41 (OV (x21 x4 )OV (x1 ) + OV (x1 )OV (x31 x4 ))
OV (x21 x2 x3 ) ∈ G
OV (x21 x2 x4 ) ∈ G
OV (x21 x3 x4 ) = OA4 (x21 x2 x3 ) − OV (x21 x2 x3 ) − OV (x21 x2 x4 )
OV (x31 x22 ) ∈ G
OV (x31 x23 ) ∈ G
OV (x31 x24 ) = OA4 (x31 x22 ) − OV (x31 x22 ) − OV (x31 x23 )
46
OV (x21 x22 x3 ) = OV (x21 x22 )OV (x1 ) − OV (x31 x2 )
OV (x21 x2 x23 ) = OV (x21 x23 )OV (x1 ) − OV (x31 x22 )
OV (x21 x2 x24 ) = OV (x21 x24 )OV (x1 ) − OV (x31 x24 )
OV (x31 x2 x3 ) = OV (x21 x2 x3 )OV (x1 ) + OV (x21 x22 x3 ) − OV (x21 x2 x23 ) − OV (x21 x2 x3 x4 )
OV (x31 x2 x4 ) = OV (x21 x2 x4 )OV (x1 ) + OV (x21 x22 x4 ) + OV (x21 x2 x3 x4 ) − OV (x21 x2 x24 )
OV (x31 x3 x4 ) = OV (x21 x3 x4 )OV (x1 ) − OV (x21 x2 x3 x4 ) + OV (x21 x23 x4 ) − OV (x21 x3 x24 )
OV (x41 x22 ) = 12 (OV (x31 x22 )OV (x1 ) + OV (x1 )OV (x31 x22 ))
OV (x41 x23 ) = 21 (OV (x31 x23 )OV (x1 ) + OV (x1 )OV (x31 x23 ))
OV (x41 x24 ) = 21 (OV (x31 x24 )OV (x1 ) + OV (x1 )OV (x31 x24 ))
OV (x31 x22 x3 ) = 21 (OV (x31 x22 )OV (x1 ) − OV (x41 x22 ) − OV (x1 x2 x3 )OV (x21 x2 ) − OV (x31 x2 x3 x4 ))
OV (x31 x22 x4 ) = 21 (OV (x31 x22 )OV (x1 ) − OV (x41 x22 ) + OV (x1 x2 x3 )OV (x21 x2 ) − OV (x31 x2 x3 x4 ))
OV (x31 x2 x23 ) = 21 (OV (x31 x23 )OV (x1 ) − OV (x41 x23 ) − OV (x1 x2 x3 )OV (x21 x3 ) − OV (x31 x2 x3 x4 ))
OV (x31 x23 x4 ) = 21 (OV (x31 x23 )OV (x1 ) − OV (x41 x23 ) + OV (x1 x2 x3 )OV (x21 x3 ) − OV (x31 x2 x3 x4 ))
OV (x31 x2 x24 ) = 21 (OV (x31 x24 )OV (x1 ) − OV (x41 x24 ) − OV (x1 x2 x3 )OV (x21 x4 ) + OV (x31 x2 x3 x4 ))
OV (x31 x3 x24 ) = 21 (OV (x31 x24 )OV (x1 ) − OV (x41 x24 ) + OV (x1 x2 x3 )OV (x21 x4 ) + OV (x31 x2 x3 x4 ))
47
Vita
J. Cameron Atkins
Date of Birth:
March 23, 1988
Place of Birth: Richmond, Virginia
Education:
• Master of Arts in Mathematics,
Wake Forest University, expected May 2012.
Thesis title: The Invariant Theory of k−1 [x1 , x2 , . . . , xn ] under Permutation Representations
Advisor: Ellen Kirkman
• Bachelor of Science in Mathematics,
James Madison Univeristy, May 2010.
Honors and Awards:
• Pi Mu Epsilon, Wake Forest University
Experience
• Graduate Teaching Assistantship, Wake Forest University
• Research Day, Winston-Salem, NC 2012, Poster Presentation
• MAA Conference, Petersburg, VA, 2010, Poster Presentation
• SUMS Conference, Harrisonburg, VA 2009, Poster Presentation
48