ON THE DECOMPOSITION NUMBERS OF STEINBERG’S TRIALITY
GROUPS 3 D4 (2n ) IN ODD CHARACTERISTICS
FRANK HIMSTEDT AND SHIH-CHANG HUANG
A BSTRACT. We determine the `-modular decomposition matrices of Steinberg’s triality
groups 3 D4 (2n ) for all primes ` 6= 2 except for some entries in the unipotent characters.
As an application, we classify all absolutely irreducible representations of 3 D4 (2n ) in
non-defining characteristic up to a certain degree.
1. I NTRODUCTION
The `-modular decomposition matrices of Steinberg’s triality groups 3 D4 (q) for all odd
prime powers q = pn and all odd primes ` 6= p were determined by M. Geck in [8] except
for a few entries, and he also obtained important results in the case ` = 2. Continuing
his work, the 2-modular decomposition matrices of 3 D4 (q) for odd q were computed by
the first author in [12], also leaving some parameters in the decomposition numbers of the
unipotent characters.
In this article, we treat the remaining case in non-defining characteristic, that is, we
calculate the `-modular decomposition matrices of 3 D4 (2n ), n > 0, for all odd primes `.
It turns out that the decomposition matrices of 3 D4 (2n ) in non-defining characteristic essentially coincide with the decomposition matrices of 3 D4 (q) for odd q. However, there
are a few decomposition numbers in the unipotent characters which we are not able to determine, and of course, the decomposition matrices of 3 D4 (2n ) might differ from those
of 3 D4 (q) for odd q in these unknown entries. The unknown entries are multiplicities of
cuspidal unipotent Brauer characters in the reduction modulo ` of ordinary characters of
large degree. It seems to be necessary to use methods different from the ones in this paper,
like module theoretic arguments as in [22], [27], [28], [32], to get more information on the
unknown entries.
The techniques we use to determine the decomposition matrices are similar to those
in [8], [9], [12], [13]. With the help of CHEVIE [10], we compute scalar products of projective characters with the ordinary unipotent irreducible characters of 3 D4 (2n ) to get an
approximation of the decomposition matrix of the unipotent blocks. Some of these projective characters are Harish-Chandra induced Gelfand-Graev characters, others are constructed by inducing defect 0 characters of proper parabolic subgroups of 3 D4 (2n ) using
the generic character tables in [11]. Additionally, we use Hecke algebra methods and relations which are obtained by expressing the reduction modulo ` of ordinary characters as
linear combinations of basic sets of Brauer characters. The decomposition matrices of the
non-unipotent blocks are derived from Bonnafé’s and Rouquier’s modular version of the
Jordan decomposition of characters.
Date: April 23, 2011.
1
2
FRANK HIMSTEDT AND SHIH-CHANG HUANG
Similar to [12], we use the decomposition matrices to classify all absolutely irreducible
representations of the groups 3 D4 (2n ) in non-defining characteristics up to a certain degree, solving a problem proposed by P.H. Tiep and A.E. Zalesskii (see [30, Problem 1.3]
and [31, Problem 5.1]) in the special case of the triality groups.
This paper is organized as follows: In Section 2, we fix notation and describe the general setup. Section 3 contains the main results, the `-modular decomposition matrices of
the triality groups 3 D4 (2n ) for odd `, which are proved in Section 4. In Section 5 we consider modular representations of 3 D4 (2n ) in odd characteristics of relatively small degree.
Decomposition matrices and degrees of Brauer characters are given in two Appendices.
2. N OTATION AND SETUP
We choose the notation similar to that in [8] and [11]. In particular, let Φ be a root
system of type D4 in some Euclidean space V , with basis ∆ = {r1 , r2 , r3 , r4 } of simple
roots such that r1 , r3 and r4 are orthogonal to each other. We fix a simple adjoint linear
algebraic group G with root system Φ defined over a finite field Fq as described in [8], with
the only difference that q = 2n is now a power of 2. Let F : G → G be the Frobenius
map in [8] which is composed of a field automorphism and a graph automorphism induced
by the symmetry r1 7→ r3 , r3 7→ r4 , r4 7→ r1 , r2 7→ r2 of Φ, so that G := GF = 3 D4 (q).
Let T be a maximally split torus of G, contained in a Borel subgroup B of G, and let P ,
Q be maximal parabolic subgroups of G containing B as in [11]. The long-root parabolic
subgroup P has Levi factor LP ∼
= SL2 (q 3 ) × Zq−1 , the short-root parabolic subgroup Q
∼
has Levi factor LQ = SL2 (q) × Zq3 −1 and we have
|B| = q 12 φ21 φ3 , |P | = q 12 φ21 φ2 φ3 φ6 , |Q| = q 12 φ21 φ2 φ3 , |G| = q 12 φ21 φ22 φ23 φ26 φ12 ,
where φ1 = q − 1, φ2 = q + 1, φ3 = q 2 + q + 1, φ6 = q 2 − q + 1, φ12 = q 4 − q 2 + 1.
Conjugacy classes and class fusions of B, P and Q are given in [11].
Let ` be an odd prime and (K, R, k) an `-modular splitting system for all subgroups
of G. By a character we mean an ordinary character afforded by a representation over K.
Brauer characters, blocks, decomposition numbers will always be taken with respect to `.
We write ϑ̆ for the restriction of a class function ϑ of G to the set of `-regular elements.
For subgroups H of G we write Irr(H) for the set of ordinary irreducible characters
of H. If χ1 , χ are characters of subgroups H1 ⊆ H of G, respectively, we write χH
1
for the induced character and χH1 for the restriction to H1 . We use the same notation
for the irreducible characters of B, P , Q, G as [4], [11], [29]. In particular, we denote
the irreducible unipotent characters of G by 1, [ε1 ], [ρ1 ], [ρ2 ], 3 D4 [−1], 3 D4 [1], [ε2 ], St.
By [3, p. 478], the characters 3 D4 [−1], 3 D4 [1] are cuspidal, the remaining six characters
are in the principal series. The set of unipotent irreducible characters of G is partitioned
into certain subsets, called families; see [3, Section 12.3]. In the case G = 3 D4 (q), these
families are {1}, {[ε1 ]}, {[ρ1 ], [ρ2 ], 3 D4 [−1], 3 D4 [1]}, {[ε2 ]}, {St}; see [25, 1.17].
∗
Let (G∗ , F ∗ ) be dual to (G, F ). Then G∗ := G∗F is isomorphic to G by [3, p. 40].
∗
Let E(G, s) be the Lusztig series of characters of G corresponding
S to a semisimple s ∈ G .
0
∗
For each semisimple ` -element s ∈ G the set E` (G, s) := t∈CG∗ (s)` E(G, st), where
CG∗ (s)` is the set of elements of `-power order of the centralizer CG∗ (s), is a union of
blocks of G; see [2, Theorem 9.12]. The unipotent blocks of G are the blocks in E(G, 1).
Furthermore, for each semisimple `0 -element s ∈ G∗ , the set {χ̆ | χ ∈ E(G, s)} is a basic
set of Brauer characters in E` (G, s), which means that every Brauer character in E` (G, s)
can be written uniquely as a linear combination with integer coefficients of the Brauer
characters χ̆, χ ∈ E(G, s); see for example [2, Theorem 14.4].
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
3
3. D ECOMPOSITION MATRICES
In this section we describe the `-modular decomposition numbers of G = 3 D4 (q),
q = 2n , for all odd primes ` dividing the group order |G| = q 12 φ21 φ22 φ23 φ26 φ12 . The
methods we use and the presentation are similar to [12], [13], inspired by [8], [9], [21].
The decomposition matrices of G depend on which of the factors φi are divisible by `. In
the “generic” case ` > 3, the prime ` divides exactly one of φ1 , φ2 , φ3 , φ6 , φ12 . If ` = 3,
then we have either ` | φ1 , φ3 or ` | φ2 , φ6 .
We start with some comments on the decomposition matrices of the unipotent blocks.
The first column of Tables A.1-A.6 in Appendix A gives notation for the irreducible ordinary characters, the first row fixes notation for the irreducible Brauer characters in the
unipotent blocks. The second row describes the modular Harish-Chandra series of the irreducible Brauer characters; see [9] for background on modular Harish-Chandra theory.
Columns labeled by “ps” correspond to Brauer characters in the principal series, columns
labeled by “c” belong to cuspidal Brauer characters. The Levi subgroup LP has a cuspidal
unipotent irreducible Brauer character if and only if ` | q 3 + 1, namely the modular Steinberg character ϕStP . This follows for example from the decomposition numbers of the
unipotent characters of GL2 (q 3 ), which are well-known, and Lemma 4.3 and Theorem 6.3
e1 for the corresponding modular Harish-Chandra
in [15]. If ϕStP is cuspidal, we write A
series of G. Similarly, LQ has a cuspidal unipotent irreducible Brauer character if and only
if ` | q + 1, namely the modular Steinberg character ϕStQ . If ϕStQ is cuspidal, we denote
the corresponding modular Harish-Chandra series of G by A1 . The following theorem will
be proved in Section 4. In the decomposition matrices mentioned in the theorem and the
table of scalar products in Appendix A, zeros are replaced by dots.
Theorem 3.1. Let ` be an odd prime dividing the order of G = 3 D4 (q), q = 2n , n > 0.
(a) If ` - q 4 − q 2 + 1, then the `-modular decomposition matrix of the unipotent blocks
of G is given by Tables A.1-A.6 in the Appendix. The occurring parameters satisfy
the following bounds:
(i) If 3 6= ` | q + 1:
1 ≤ a, b ≤ q/2. If ` 6= 5 or n ≡ 10 mod 20, then a, b ≥ 2.
(ii) If 3 6= ` | q 2 + q + 1:
1 ≤ a ≤ q and 2a − 3 ≤ b ≤ (q 2 − q)/2.
2 ≤ c ≤ q/2 if q > 2, and c = 2 if q = 2.
(iii) If 3 6= ` | q 2 − q + 1:
2 ≤ b ≤ (q 2 − q)/2 and 0 ≤ c ≤ q/2 and 0 ≤ d ≤ (q − 2)/2.
(iv) If ` = 3 | q − 1:
1 ≤ a ≤ q and a − 1 ≤ b ≤ (q 2 − q)/2 and 1 ≤ c ≤ q − 1.
(v) If ` = 3 | q + 1:
0 ≤ a ≤ (q 2 − q)/2 and 1 ≤ d ≤ q/2.
a + 1 ≤ b ≤ q 3 /2. If n ≡ 3 mod 6, then b ≥ a + 2.
1 ≤ c ≤ q/2. If n ≡ 3 mod 6, then c ≥ 2.
(b) If ` | q 4 − q 2 + 1, then the principal block of G is cyclic with Brauer tree
1
s
[ρ1 ]
s
St
s
{χ14 }
sg
3
D4 [−1]
s
4
FRANK HIMSTEDT AND SHIH-CHANG HUANG
The remaining unipotent irreducible characters of G have defect 0.
(c) The `-modular decomposition matrices of the non-unipotent blocks of G coincide
with the matrices for odd q given in [8, (6.2) - (6.8)], where the parameter a
occurring in χ9,St satisfies: a = 2 if ` divides q + 1 and a = 0 otherwise.
Remarks:
(a) Comparing Theorem 3.1 with M. Geck’s results [8], one can see that those decomposition numbers of G which we were able to determine for even q essentially
coincide with those for odd q. In particular, the Brauer tree in Theorem 3.1 (b) is
the same as the one for odd q in [8]. A possible explanation for this phenomenon
is given in the third paragraph of Section 4. For the convenience of the reader and
since we provide additional information on the modular Harish-Chandra series,
we include the decomposition matrices for the unipotent blocks in Appendix A.
(b) In the case 3 6= ` | q − 1, the prime ` is linear for G in the sense of [14, Section 6]
and the decomposition numbers and modular Harish-Chandra series in Table A.1
follow from results of G. Hiß; see [14, Theorem 6.3.7] and [15, Lemma 4.3].
(c) Together with the ordinary character table of G, which was determined by N. Spaltenstein, D.I. Deriziotis and G.O. Michler in [4], [29] the decomposition matrices
in Theorem 3.1 determine all irreducible `-modular Brauer characters for ` > 2
except for at most two.
(d) Theorem 3.1 confirms [9, Conjecture 3.4] in the special case G = 3 D4 (2n ). In
particular, the decomposition matrices in Appendix A are unitriangular and for
each cuspidal unipotent χ ∈ Irr(G), the Brauer character χ̆ is irreducible.
(e) All decomposition numbers of 3 D4 (2) are known; see the GAP [6] library and the
Atlas of Brauer characters [20].
4. P ROOFS : D ECOMPOSITION M ATRICES
This section contains the proof of Theorem 3.1. The methods we use are similar to those
in [8], [12] and [13]. Our main tool are projective characters, that is, ordinary characters
of projective kG-modules. Due to Brauer reciprocity, the decomposition numbers of G are
given by the scalar products of the ordinary irreducible characters of G with the ordinary
characters corresponding to projective indecomposable kG-modules.
The scalar products we need are listed in Table A.7 in Appendix A. The entries of this
table are the scalar products (χ, ψ), where χ runs through the ordinary unipotent irreducible characters of G and ψ runs through the characters specified in the first row of the
table. The characters ΓT , ΓLP , ΓLQ , ΓG are the Gelfand-Graev characters of the Levi subG
groups T , LP , LQ and G, respectively, and RL
denotes Harish-Chandra induction. The
G
G
G
characters RT (ΓT ), RLP (ΓLP ), RLQ (ΓLQ ) and ΓG are projective for all odd primes `,
since a Gelfand-Graev character is induced from an `0 -subgroup and Harish-Chandra induction preserves projectives; see [15, p. 220]. The remaining characters in the first row
of Table A.7 are of the form ψ G , where ψ is a character of a proper parabolic subgroup
of G. In general, ψ G is not projective for all odd primes `. However, we use ψ G only for
those primes `, where ψ has `-defect 0. The scalar products involving the Harish-Chandra
induced Gelfand-Graev characters can be determined as described in [8]. The values and
scalar products of the characters which are induced from parabolic subgroups can easily
be computed using CHEVIE [10] and the character tables and class fusions in [11].
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
5
It might be possible that the characters in Table A.7 which are induced from parabolic
subgroups could also be obtained by translating Geck’s construction of the generalized
Gelfand-Graev characters and their modifications (see [7, Sections 3.10-3.13]) to even q
and that the slight differences between the character tables of G and B for odd q and even q
do not affect the scalar products with the unipotent characters. But we did not check this
in detail since it was easier to use the CHEVIE tables from [11].
Another essential tool in the proof of Theorem 3.1 are the decomposition numbers of the
G
G
Hecke algebra H := EndRG (RB
), where RB
is the RG-permutation module on the cosets
of the Borel subgroup B. By R. Dipper’s result [5, Corollary 4.10] the decomposition
matrix of H is a submatrix of the decomposition matrix of E` (G, 1). M. Geck’s [8] construction of irreducible representations ind, sgn, σ1 , σ2 , S1 , S−1 of H for odd q is also valid
for even q. These representations afford irreducible representations of HK := K ⊗R H (by
extending scalars), which we will also denote by ind, sgn, σ1 , σ2 , S1 , S−1 , respectively.
There is a natural bijection (“Fitting correspondence”) between the (isomorphism classes
of) irreducible representations of HK and the set of irreducible constituents of the ordinary
permutation character 1G
B . As in the odd case [8, Section 5], the representations ind, sgn,
σ1 , σ2 , S1 , S−1 correspond to 1, St, [ε2 ], [ε1 ], [ρ1 ], [ρ2 ], respectively.
Proposition 4.1 (M. Geck, [8]). For each R-representation S of H, let S be the representation of Hk := k ⊗R H which is obtained by reduction modulo `. Then:
(i) The representations ind, sgn, σ 1 , σ 2 are irreducible. If ` - q 3 + 1, they are all
distinct. If ` | q + 1, they are all equal. For 3 6= ` | q 2 − q + 1 we have ind = σ 1
and sgn = σ 2 .
(ii) S 1 is reducible if and only if ` | q 4 − q 2 + 1 or ` | q 2 + q + 1. If ` | q 2 + q + 1,
then S 1 has the two constituents σ 1 and σ 2 . If ` | q 4 − q 2 + 1, then S 1 has the two
constituents ind and sgn.
(iii) S −1 is reducible if and only if ` | q 2 + q + 1 or ` | q 2 − q + 1. In this case, S −1
has the two constituents ind and sgn.
(iv) If S 1 , S −1 are both irreducible, then they are not isomorphic to each other.
Proof. The proof of [8, Proposition 5.1] also holds for even q.
A further important ingredient in the computation of the decomposition numbers of the
unipotent blocks are relations with respect to the basic set of unipotent characters. More
precisely, let ψ ∈ E` (G, 1) be a non-unipotent ordinary character. As described at the end
of Section 2, the set {χ̆ | χ ∈ E(G, s)} is a basic set of Brauer characters in E` (G, s), so
that there are unique a, b, . . . , h ∈ Z such that
(1) ψ̆ = a · 1 + b · [ε1 ] + c ·[ρ1 ] + d·[ρ2 ] + e· 3 D4 [−1] + f · 3 D4 [1] + g ·[ε2 ] + h·St,
where we omitted the “ ˘ ” on the right hand side for better readability. We call (1) a
relation with respect to the basic set of unipotent characters. Relations can be used to
prove lower bounds for decomposition numbers. For all odd `, the relations with respect to
the basic set of unipotent characters can easily be computed using the character table of G
in CHEVIE [10].
4.1. Proof of Theorem 3.1, non-unipotent blocks. We begin with the determination of
the decomposition numbers of the non-unipotent blocks, that is, with the proof of part (c)
of the theorem. Let ` be an odd prime and s ∈ G∗ , s 6= 1 a semisimple `0 -element.
Since G is of adjoint type, the center Z(G) = {1} is connected, and since the order of s is
not divisible by 2 (the only bad prime for a root system of type D4 ), we can apply C. Bonnafé’s and R. Rouquier’s modular version of the Jordan decomposition of characters [1,
6
FRANK HIMSTEDT AND SHIH-CHANG HUANG
Theorem 11.8]. Hence, Lusztig induction induces a 1-1-correspondence between the sets
E` (G, s) and E` (CG∗ (s), 1) of ordinary irreducible characters, and with respect to this correspondence the decomposition matrices of E` (G, s) and E` (CG∗ (s), 1) coincide (after a
suitable ordering of the columns).
The Dynkin type of CG∗ (s) is given in [4, Table 2.2a]. If s 6= 1 is not of type s9 ,
then the Dynkin type of CG∗ (s) is A2 , A1 or A0 and so the decomposition numbers of
E` (CG∗ (s), 1) are known; see [19]. If s is of type s9 , then we see from [4, Table 2.2a] that
there is a cyclic central subgroup Sσ of CG∗ (s) such that CG∗ (s)/Sσ ∼
= PGU3 (q); note
that SU3 (q) ∼
= PGU3 (q) if q 6≡ −1 mod 3. The decomposition numbers of the unipotent
irreducible characters of PGU3 (q) follow from G. Hiß’ Theorem 4.1 in [16] describing the
structure of the permutation module of PGU3 (q) on the cosets of its Borel subgroup. This
completes the proof of part (c) of the theorem.
4.2. Proof of Theorem 3.1, unipotent blocks. We proceed analogously to [13, 5.7]:
Using [4, Table 2.1] and [11, Table A.1], we count the elements of `-power order in G∗ to
get the numbers in the right most column of Tables A.1-A.6; see also the information in the
character sum procedures of the character table of G in CHEVIE [10]. Some of the numbers are zero for certain values of n: The ordinary irreducible characters corresponding to
the
•
•
•
•
last row of Table A.1 exist if and only if ` 6= 5 or n ≡ 0 mod 20,
last row of Table A.2 exist if and only if ` 6= 5 or n ≡ 10 mod 20,
last row of Table A.5 exist if and only if n ≡ 0 mod 6,
last row of Table A.6 exist if and only if n ≡ 3 mod 6,
where q = 2n . The general strategy to determine the decomposition numbers of the unipotent blocks is as follows: We use the scalar products of those characters in Table A.7 which
are projective to obtain an approximation of the decomposition matrix of E` (G, 1). As
in [13], this already gives us the unitriangular shape of the decomposition matrix. Then,
using Proposition 4.1 and relations with respect to the basic set of unipotent characters, we
obtain most of the decomposition numbers as well as upper and lower bounds for those
numbers which we cannot determine. Finally, in some cases, we can improve these bounds
using some further projective characters from Table A.7.
Case 3 6= ` | q −1: By [14, Proposition 6.3.4], ` is a linear prime for G, and the decomposition numbers in Table A.1 follow from [14, Theorem 6.3.7]. The modular Harish-Chandra
series can be concluded from Proposition 4.1 and [15, Lemma 4.3].
We now describe the most complicated case ` = 3 | q + 1 in detail. The determination
of the decomposition numbers is similar to M. Geck’s approach in [8].
Case ` = 3 | q + 1: Since Q χ8 , B χ9 , P χ9 , P χ11 (1) have defect 0, the characters RTG (ΓT ),
G
G
G
G
G
RL
(ΓLQ ), RL
(ΓLP ), Q χG
8 , B χ9 , P χ9 , P χ11 (1) , ΓG in Table A.7 are projective.
Q
P
Using the scalar products with these character and the fact that [ρ1 ] has defect 0, we get the
approximation in Table 1 of that part of the decomposition matrix of G which corresponds
to the unipotent characters.
Let φ1 , . . . , φ8 be the irreducible Brauer character in E` (G, 1) so that φi corresponds to
the i-th column of the approximated decomposition matrix, and let Φi denote the character
of the projective indecomposable module corresponding to φi . We set a := ([ε2 ], Φ5 ),
b := (St, Φ5 ), c := (St, Φ6 ) and d := (St, Φ7 ).
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
7
TABLE 1. Approximation to the decomposition matrix of the unipotent
characters in case ` = 3 | q + 1.
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
1
1
.
2
.
.
1
1
.
1
.
1
.
.
.
1
.
.
1
.
.
.
.
.
.
.
.
1
.
.
1
q 2 +q+2
2
.
.
.
.
1
.
q 2 −q
2
q3
2
.
.
.
.
.
1
.
q
2
.
.
.
.
.
.
1
q−1
.
.
.
.
.
.
.
1
From the above approximation and Proposition 4.1 we already get all decomposition
numbers in Φ1 , Φ3 , Φ8 and the upper bounds for a, b, c. Furthermore, φ1 and φ3 are the
G
only irreducible Brauer characters in the principal series. Since RL
(ΓLP ) − [ρ1 ] is the
P
character of a projective module containing Φ4 , we get (St, Φ4 ) ≤ 1. From the relations
corresponding to χ9,qs0 , χ7,St and χ10,St we obtain ([ρ2 ], Φ2 ) = (St, Φ2 ) = ([ε2 ], Φ4 ) =
(St, Φ4 ) = 1, proving all assertions about the decomposition numbers in Φ2 and Φ4 . By
construction, φ2 and φ4 are the only irreducible Brauer characters in the modular Harishe1 , respectively. So φ5 , φ6 , φ7 , φ8 are cuspidal. The lower bounds
Chandra series A1 and A
for b, c, d follow from the relations corresponding to χ7,St and χ15 .
The character Q χG
9 − [ρ1 ] is the character of a projective module because Q χ9 and [ρ1 ]
have defect 0. Therefore Q χG
9 − [ρ1 ] = (q + 1) · Φ7 + A · Φ8 + Ψ, where A is a nonnegative
integer and Ψ is a sum of characters of projective modules in non-unipotent blocks. From
1
Table A.7 we get (q + 1)d ≤ (q 2 + q + 2)/2 and so d ≤ 2q + q+1
, which implies d ≤ q/2,
completing the proof in the case ` = 3 | q + 1.
The remaining cases are analogous to the case ` = 3 | q + 1. We only give the projective
characters we used in each single case.
G
G
(ΓLQ ), RL
Case 3 6= ` | q+1: Here we use the projective characters RTG (ΓT ), RL
(ΓLP ),
Q
P
G
G
G
G
χ
,
χ
,
χ
,
χ
,
Γ
to
obtain
an
approximation
of
the
decomposition
matrix.
P 13 P 12 P 9 Q 7
G
G
G
(ΓLQ ), RL
Case 3 6= ` | q 2 + q + 1: Use the projective characters RTG (ΓT ), RL
(ΓLP ),
Q
P
G
G
G
G
,
χ
,
χ
,
χ
,
Γ
to
obtain
an
approximation
of
the
decomposition
matrix.
The
χ
G
P 13 P 12 Q 12 Q 7
G
upper bound for c can finally be improved using Q χ9 .
G
G
Case 3 6= ` | q 2 − q + 1: Use the projective characters RTG (ΓT ), RL
(ΓLQ ), RL
(ΓLP ),
Q
P
G
G
G
G
Q χ8 , Q χ11 , P χ9 , P χ11 (1) , ΓG to obtain an approximation of the decomposition matrix.
The upper bound for d can finally be improved using Q χG
12 .
G
G
Case ` = 3 | q − 1: Use the projective characters RTG (ΓT ), RL
(ΓLQ ), RL
(ΓLP ), P χG
13 ,
Q
P
G
G
G
χ
,
χ
,
χ
,
Γ
to
obtain
an
approximation
of
the
decomposition
matrix.
The
upper
P 12 Q 12 Q 7
G
bound for c can finally be improved using B χG
10 .
Case ` | q 4 − q 2 + 1: Ignoring the characters of G of defect 0 and using the projective
G
characters RTG (ΓT ), RL
(ΓLQ ), P χG
12 , ΓG one can obtain an approximation to the deQ
composition matrix. Then, the Brauer tree in part (b) of the theorem follows from general
properties of the decomposition matrices of cyclic blocks; see [17, Theorem 2.1.5].
8
FRANK HIMSTEDT AND SHIH-CHANG HUANG
5. I RREDUCIBLE REPRESENTATIONS OF RELATIVELY SMALL DEGREE
In the same way as in [12] we apply the decomposition matrices to obtain information on
modular representations of 3 D4 (2n ) of relatively small degree. More specifically, we are
interested in the following version of a problem formulated by P.H. Tiep and A.E. Zalesskii
(see [30, Problem 1.3] and [31, Problem 5.1]):
For a finite quasisimple group G, an algebraically closed field k of characteristic ` ≥ 0
and ε > 0, classify all nontrivial irreducible kG-representations of degree < d` (G)2−ε ,
where d` (G) is the smallest degree of a nonlinear kG-representation.
See [30] and [31] for comments and applications related to the above problem. In
the following, we consider the special case G = 3 D4 (q) for q even. Since the ordinary
character tables of these groups are known (see [4], [10], [29]) and the case of defining
characteristic ` = 2 follows from far-reaching results of F. Lübeck [23], we only have to
consider representations over fields of non-defining characteristic ` 6= 2. It was shown by
K. Magaard, G. Malle and P.H. Tiep [26, Section 4] that d` (G) ≥ q 5 − q 3 + q − 1 and that
equality holds if ` | q + 1. It turns out that for even q one has
(
q 5 − q 3 + q − 1 if ` | q + 1,
d` (G) =
q5 − q3 + q
else,
in close analogy with F. Lübeck’s result [24, 4.4] for odd q. It should be mentioned that
the representations of degree ≤ 250 of G (in fact, of all quasisimple finite groups) have
already been determined by G. Hiß and G. Malle [18] and that the tables of `-modular
Brauer characters of 3 D4 (2) for all primes ` can be found in the GAP [6] library or the
Atlas of Brauer characters [20]. The next theorem solves Tiep’s and Zalesskii’s problem
for G = 3 D4 (q) for q ≥ 4 even. So together with the results just mentioned and [12], this
solves Tiep’s and Zalesskii’s problem for 3 D4 (q) for all prime powers q.
Theorem 5.1. Table B.1 shows the degrees deg(ρ), the Brauer characters and the multiplicities of all nontrivial absolutely irreducible representations ρ of G for even q ≥ 4 over
fields of characteristic ` 6= 2 satisfying the condition deg(ρ) < d` (G)2 .
Table B.1 is the analogue of [12, Table A.14]. The first column of Table B.1 is a list of all
integers t such that there is an absolutely irreducible representation of G = 3 D4 (q), q ≥ 4
even, of degree 1 < t < d` (G)2 over a field of characteristic ` 6= 2. The second column
describes the characteristics of the fields over which the representations are defined. The
third and fourth columns give notation and the number of the Brauer characters of these
representations.
For fixed q, the degrees in the first column are given in strictly increasing order (for
this reason the case q = 2 is excluded from Table B.1). By “` - x” in the second column
we mean that a representation of this degree exists for all characteristics not dividing x,
including characteristic 0, and “` - q; x no power of `” means that a representation of this
degree exists for all characteristics 6= 2, including characteristic 0, but excluding those `
where x is a power of `. The notation for the unipotent Brauer characters in the third
column is the same as in Tables A.1-A.6; except for ` | q 4 − q 2 + 1 where we use a notation
which should be clear from the Brauer tree in Theorem 3.1. The non-unipotent Brauer
characters in Table B.1 are φi,1 := χ̆i,1 for i = 3, 4, 7, 9. Note that in the second column φi
denotes a cyclotomic polynomial in q while in the third column φi is the notation for a
Brauer character. In the fourth column, we write m`0 for the largest divisor of an integer m
which is prime to `.
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
9
Table B.2 is a list of all integers t ≤ 10000 such that there is a power q of some prime p
and a nontrivial absolutely irreducible representation of degree t of 3 D4 (q) over a field of
characteristic 6= p. The columns labeled by “Characteristics `” describe the characteristics
of the fields over which the respective representations are defined. As above, by “` - x”
we mean that a representation of this degree exists for all characteristics not dividing x
(including characteristic 0). The data in Table B.2 follows from [12, Table A.15] for odd q,
Table B.1 for even q ≥ 4 and the GAP [6] library or [20] for q = 2.
Proof. (of Theorem 5.1) We use the same techniques as F. Lübeck in [24]; see also the
proof of [12, Theorem 5.1]. The decomposition matrices in Theorem 3.1 determine the
degrees of all irreducible Brauer characters of G except for possibly deg(φ7 ) and deg(φ8 ).
The upper bounds for the unknown decomposition number a (if there is such an unknown
decomposition number) lead to a lower bound for deg(φ7 ), and the lower bounds for a and
the upper bounds for the unknown decomposition numbers in St lead to a lower bound for
deg(φ8 ). These lower bounds imply deg(φ7 ), deg(φ8 ) ≥ deg(φ2 )2 ≥ d` (G)2 .
ACKNOWLEDGMENTS
Part of this work was done during visits of the first author at the Department of Mathematics of the Chiba University, Japan, and of the National Cheng Kung University, Taiwan.
He thanks all persons of these departments for their hospitality and the Japan Society for
the Promotion of Science (JSPS) and the National Center for Theoretical Sciences (South)
for supporting his visits. The second author acknowledges the support of his research from
the National Science Council, ROC. We thank F. Lübeck for providing a GAP program to
compare rational numbers which are given as polynomials in q.
A PPENDIX A: D ECOMPOSITION NUMBERS OF UNIPOTENT CHARACTERS
TABLE A.1. Decomposition numbers of E` (G, 1) for 3 6= ` | q − 1. In
the right most column, `f is the largest power of ` dividing q − 1.
Char.
φ1
φ2
φ3
φ4
φ5
φ6
φ7
φ8
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ3,1 }
{χ3,St }
{χ5,1 }
{χ5,St }
{χ6 }
ps
1
.
.
.
.
.
.
.
1
.
1
.
1
ps
.
1
.
.
.
.
.
.
1
.
.
1
1
ps
.
.
1
.
.
.
.
.
1
1
1
1
2
ps
.
.
.
1
.
.
.
.
1
1
1
1
2
c
.
.
.
.
1
.
.
.
.
.
.
.
.
c
.
.
.
.
.
1
.
.
.
.
.
.
.
ps
.
.
.
.
.
.
1
.
.
1
1
.
1
ps
.
.
.
.
.
.
.
1
.
1
.
1
1
Number
1
1
1
1
1
1
1
1
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1
f
f
12 (` − 1)(`
− 5)
10
FRANK HIMSTEDT AND SHIH-CHANG HUANG
TABLE A.2. Decomposition numbers of E` (G, 1) for 3 6= ` | q + 1. In
the right most column, `f is the largest power of ` dividing q + 1.
Char.
φ1
φ2
φ3
φ4
φ5
φ6
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ7,1 }
{χ7,St }
{χ10,1 }
{χ10,St }
{χ15 }
ps
1
1
.
.
.
.
1
1
.
.
.
.
.
A1
.
1
.
.
.
.
.
1
1
1
.
.
.
ps
.
.
1
.
.
.
.
.
.
.
.
.
.
ps
.
.
.
1
.
.
.
.
.
.
.
.
.
c
.
.
.
.
1
.
.
a
1
a−1
1
a−1
a−2
c
.
.
.
.
.
1
.
b
1
b−1
1
b−1
b−2
φ7
e1
A
.
.
.
.
.
.
1
1
.
.
1
1
.
φ8
Number
c
.
.
.
.
.
.
.
1
.
1
.
1
1
1
1
1
1
1
1
1
1
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1
f
f
12 (` − 1)(`
− 5)
TABLE A.3. Decomposition numbers of E` (G, 1) for 3 6= ` | q 2 + q + 1.
In the right most column, `f is the largest power of ` dividing q 2 + q + 1.
Char.
φ1
φ2
φ3
φ4
φ5
φ6
φ7
φ8
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ4,1 }
{χ4,qs }
{χ4,St }
{χ12 }
ps
1
.
.
1
.
.
.
.
1
1
.
.
ps
.
1
1
.
.
.
.
.
.
1
1
.
ps
.
.
1
.
.
.
1
.
1
1
.
.
ps
.
.
.
1
.
.
.
1
.
1
1
.
c
.
.
.
.
1
.
.
.
.
.
.
.
c
.
.
.
.
.
1
a
b
1
a−1
b−a+1
b − 2a + 3
c
.
.
.
.
.
.
1
c
.
1
c−1
c−2
c
.
.
.
.
.
.
.
1
.
.
1
1
Number
1
1
1
1
1
1
1
1
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1
f
f
24 (` − 1)(`
− 3)
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
11
TABLE A.4. Decomposition numbers of E` (G, 1) for 3 6= ` | q 2 − q + 1.
In the right most column, `f is the largest power of ` dividing q 2 − q + 1.
Char.
φ1
φ2
φ3
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ9,1 }
{χ9,qs0 }
{χ9,St }
{χ13 }
ps
1
.
.
1
.
.
1
.
.
.
.
.
ps
.
1
.
1
.
.
.
1
.
.
.
.
ps
.
.
1
.
.
.
.
.
.
.
.
.
φ4
e1
A
.
.
.
1
.
.
1
1
1
.
1
.
φ5
φ6
φ7
φ8
c
.
.
.
.
1
.
.
b
1
.
b−1
b−2
c
.
.
.
.
.
1
.
c
.
1
c
c+1
c
.
.
.
.
.
.
1
d
.
1
d+1
d+2
c
.
.
.
.
.
.
.
1
.
.
1
1
Number
1
1
1
1
1
1
1
1
1 f
2 (` − 1)
1 f
2 (` − 1)
1 f
2 (` − 1)
1
f
f
24 (` − 1)(`
− 3)
TABLE A.5. Decomposition numbers of E` (G, 1) for ` = 3 | q − 1. In
the right most column, 3f is the largest power of 3 dividing q − 1.
Char.
φ1
φ2
φ3
φ4
φ5
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ3,1 }
{χ3,St }
χ4,1
χ4,qs
χ4,St
{χ5,1 }
{χ5,St }
{χ6 }
ps
1
.
.
1
.
.
.
.
2
1
1
1
.
2
1
3
ps
.
1
1
.
.
.
.
.
2
1
.
1
1
1
2
3
ps
.
.
1
.
.
.
1
.
1
2
1
1
.
2
1
3
ps
.
.
.
1
.
.
.
1
1
2
.
1
1
1
2
3
c
.
.
.
.
1
.
.
.
.
.
.
.
.
.
.
.
φ6
φ7
c
c
.
.
.
.
.
.
.
.
.
.
1
.
a
1
b
c
.
.
a+b
c+1
1
.
a−1
1
b+1−a c−1
a
1
b
c
a+b
c+1
φ8
c
.
.
.
.
.
.
.
1
.
1
.
.
1
.
1
1
Number
1
1
1
1
1
1
1
1
1 f
2 (3
1 f
2 (3
− 1)
− 1)
1
1
1
3 f
2 (3
3 f
2 (3
1 f
4 (3
− 1)
− 1)
− 1)(3f − 3)
12
FRANK HIMSTEDT AND SHIH-CHANG HUANG
TABLE A.6. Decomposition numbers of E` (G, 1) for ` = 3 | q + 1. In
the right most column, 3f is the largest power of 3 dividing q + 1.
Char.
φ1
φ2
φ3
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
{χ7,1 }
{χ7,St }
χ9,1
χ9,qs0
χ9,St
{χ10,1 }
{χ10,St }
{χ15 }
ps
1
1
.
2
.
.
1
1
.
.
.
.
.
.
.
.
A1
.
1
.
1
.
.
.
1
1
1
.
.
.
.
.
.
ps
.
.
1
.
.
.
.
.
.
.
.
.
.
.
.
.
φ4
e1
A
.
.
.
1
.
.
1
1
.
.
1
.
1
1
1
.
φ5
φ6
φ7
c
c
c
.
.
.
.
.
.
.
.
.
.
.
.
1
.
.
.
1
.
a
.
1
b
c
d
1
1
.
b−a−1 c−1 d−1
1
.
.
a
1
1
a+b−1
c
d+1
a+1
1
1
b−1
c−1
d
b−a−2 c−2 d−1
φ8
c
.
.
.
.
.
.
.
1
.
1
.
.
1
.
1
1
Number
1
1
1
1
1
1
1
1
1 f
2 (3
1 f
2 (3
− 1)
− 1)
1
1
1
3 f
2 (3
3 f
2 (3
1 f
4 (3
− 1)
− 1)
− 1)(3f − 3)
TABLE A.7. Scalar products of the unipotent irreducible characters with
some ordinary characters of G.
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
RTG (ΓT )
G
RL
(ΓLQ )
Q
G
RL
(ΓLP )
P
G
Q χ9
G
B χ10
G
P χ13
G
Q χ8
1
1
2
2
.
.
1
1
.
1
1
1
.
.
.
1
.
.
1
1
.
.
1
1
.
.
1
.
.
.
q+1
.
.
1
.
.
.
.
.
.
1
.
.
.
.
.
.
1
.
.
1
q
2
q 2 +q+2
2
q 2 +q+2
2
q 2 +q
2
q3
2
ON THE DECOMPOSITION NUMBERS OF 3 D4 (2n )
13
TABLE A.7. (continued)
1
[ε1 ]
[ρ1 ]
[ρ2 ]
3
D4 [−1]
3
D4 [1]
[ε2 ]
St
G
P χ12
G
Q χ11
G
B χ9
G
P χ9
G
Q χ12
G
Q χ7
G
P χ11 (1)
ΓG
.
.
.
.
1
.
.
.
.
.
.
1
.
.
.
.
.
.
1
.
.
.
.
.
.
1
.
.
.
.
.
.
1
q
q
2
q 2 −q
2
q
2
q 2 −q
2
.
.
.
.
.
.
1
q
.
.
.
.
.
.
1
q−1
.
.
.
.
.
.
.
1
q 2 −q
2
q3
2
A PPENDIX B: D EGREES OF MODULAR REPRESENTATIONS
TABLE B.1. Degrees, Brauer characters, multiplicities of all nontrivial
absolutely irreducible representations of G = 3 D4 (q), q ≥ 4 even, of
degree < d` (G)2 over fields of characteristic 6= 2.
Characteristic `
` | φ2
` - qφ2 φ12
` | φ12
` - qφ12
` | φ12
` - qφ12
` | φ12
` | φ3
` | φ12
` - qφ3 φ12
Degree
q5 − q3 + q − 1
q5 − q3 + q
1 3 6
2 q (q
− 2q 5 + 2q 3 − 2q + 1)
1 3 6
2 q (q
− 2q 3 + 1)
1
8
5
4
2
2 q(q + 2q − 2q + 3q − 2)
1 9
6
3
2 (q + 2q + q − 2)
1 3 6
3
2 q (q + 2q + 1)
1 2
3
2
2 (q − 1)(q − 2q + 2)(q + q
Brauer char.
φ2
φ2
φ[ε1 ]
φ6
φ3 D4 [1]
φ5
φχ14 ,3 D4 [−1]
φ3
φ[ρ1 ],St
φ3
Mult.
1
1
1
1
1
1
1
1
1
1
φ4
1
` | φ3
φ4
1
` - qφ3 φ6
φ4
1
+ 1)2 ` | φ6
1 9
8
6
4
3
2 (q + 2q − 2q + 2q + q −
1 3 6
5
3
2 q (q + 2q − 2q + 2q + 1)
3
2
4
2
2)
(q − 1)(q − q + 1)(q − q + 1)
`- q;
φ7,1
φ2 no power of `
(q + 1)(q 4 − q 2 + 1)(q 2 − q + 1)2
`- q;
φ4,1
φ3 no power of `
(q − 1)(q 4 − q 2 + 1)(q 2 + q + 1)2
`- q;
φ9,1
φ6 no power of `
q9 + q8 + q5 + q4 + q + 1
`- q;
φ3,1
φ1 no power of `
(q+1)`0 −1
2
(q 2 +q+1)`0 −1
2
(q 2 −q+1)`0 −1
2
(q−1)`0 −1
2
14
FRANK HIMSTEDT AND SHIH-CHANG HUANG
TABLE B.2. All degrees ≤ 10000 of nontrivial absolutely irreducible
representations of 3 D4 (pn ), p an arbitrary prime, over fields of characteristic 6= p.
Degree Group G
3
25
D4 (2)
3
26
D4 (2)
3
52
D4 (2)
3
196
D4 (2)
3
218
D4 (3)
3
219
D4 (3)
3
273
D4 (2)
3
298
D4 (2)
3
323
D4 (2)
3
324
D4 (2)
3
351
D4 (2)
3
441
D4 (2)
3
467
D4 (2)
3
468
D4 (2)
3
637
D4 (2)
3
963
D4 (4)
3
964
D4 (4)
3
1053
D4 (2)
3
1222
D4 (2)
3
1262
D4 (2)
Characteristics `
3
6= 2, 3
6= 2
6= 2
2
6= 2, 3
6= 2, 3
7
13
6= 2, 7, 13
6= 2, 7
3
7
6= 2, 3, 7
6= 2, 3
5
6= 2, 5
6= 2
3
7
Degree
1274
1664
1911
1963
2106
2184
2457
2808
3004
3005
3773
3822
3942
3969
4096
5096
6642
6643
9126
Group G
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (5)
3
D4 (5)
3
D4 (2)
3
D4 (2)
3
D4 (3)
3
D4 (2)
3
D4 (2)
3
D4 (2)
3
D4 (3)
3
D4 (3)
3
D4 (3)
Characteristics `
6= 2, 3
6= 2, 3, 7
6= 2, 3
3
6= 2, 7
6= 2, 3
6= 2, 7
6= 2, 3, 7
2, 3
6= 2, 3, 5
13
6= 2, 3
6= 3
6= 2, 13
6= 2, 3, 7, 13
6= 2, 3
2
6= 2, 3
6= 3
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