London Mathematical Society
ISSN 1461–1570
SMALL DEGREE REPRESENTATIONS OF FINITE CHEVALLEY
GROUPS IN DEFINING CHARACTERISTIC
FRANK LÜBECK
Abstract
We determine for all simple simply connected reductive linear
algebraic groups defined over a finite field all irreducible representations in their defining characteristic of degree below some bound.
These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For
large rank l our bound is proportional to l 3 and for rank 6 11 much
higher. The small rank cases are based on extensive computer calculations.
1. Introduction
In this note we give lists of projective representations of simple Chevalley groups in
their defining characteristic. There are two types of results.
First we determine for groups of rank 6 11 all such representations of degree less than
or equal some bound depending on the type (e.g., 100000 for type E8 ), see Theorem 4.4
for details. These data are produced using a collection of computer programs developed
by the author. Of course, some of the degrees given in our tables have been known before.
To give two examples: Gilkey and Seitz handle some cases for exceptional types with
computational methods in [6]. Some small rank cases are handled more theoretically like
type B2 and G2 by Jantzen in [11] (although the explicit degrees are still not too easy to
obtain from that result).
Secondly we determine for groups of classical type with rank l all representations of
degree at most l 3 /8 for type Al , and of degree at most l 3 for the other types. For large l
this is a small list, given in Theorem 5.1, and for small l this range is easily covered by our
tables mentioned above. This extends similar results by Liebeck with bounds proportional
to l 2 , see [13, 5.4.11] and the references given there.
Our results contains in particular a complement to the tables of representations in nondefining characteristic up to degree 250 worked out by Hiß and Malle in [7].
We fix some notation for the whole paper. Let G̃ be a finite twisted or non-twisted simple
Chevalley group in characteristic p (we consider 2 F(2)0 as sporadic and exclude it here,
but see the corresponding remark after Theorem 4.4).
There is an associated connected reductive simple algebraic group G over F¯ p of simply
connected type, a Frobenius endomorphism F of G, a number q = p f with 2 f ∈ Z and an
m ∈ N such that:
— G is defined over Fqm via F m .
— For the group of F-fixed points G(q) with center Z we have G̃ ∼
= G(q)/Z.
— G(q) is the quotient of the universal covering group of G̃ by the p-part of its center.
Received 25th August 2000, revised 28th May 2001; published ???.
1991 Mathematics Subject Classification 20C33, 20C44, 20G05, 20G40
c 200X, Frank Lübeck
LMS J. Comput. Math. VV (200X) 1–32
Small degree representations in defining characteristic
(Note, that this includes the Suzuki and Ree groups.)
So, asking for the projective representations of G̃ in characteristic p is the same as asking
for the representations of G(q) in characteristic p. These can be constructed by restricting
certain representations, called highest weight representations, of the algebraic group G to
G(q). This is explained in Section 2. In Section 3 we shortly describe how our computer
programs for computing weight multiplicities work. In Section 4 we describe our main
result consisting of lists of small degree representations for groups of rank at most 11. The
lists are printed in Appendix 6. Finally, in Section 5 we consider groups of larger rank.
Acknowledgements. I wish to thank Kay Magaard, Gunter Malle and Gerhard Hiß for
useful discussions about the topic of this note.
2. Representations in defining characteristic
There are several well readable introductions to this topic, for example Humphreys’
survey [9]. A detailed reference is Jantzen’s book [12]. We recall some of the basic facts.
For G of simply connected type of rank l as in the introduction let T be a maximal torus
of G, X ∼
= Zl its character group and Y ∼
= Zl its co-character group.
Let {α1 , . . . , αl } ⊂ X be a set of simple roots for G with respect to T and α∨i ∈ Y the
coroot corresponding to αi , i = 1, . . . , l. Viewing X ⊗ R as Euclidean space we define the
fundamental weights ω1 , . . . , ωl ∈ X ⊗ R as the dual basis of α∨1 , . . . , α∨l . This is a Z-basis
of X (because G is simply connected).
There is a partial ordering 6 on X defined by ω 6 ω0 if and only if ω0 − ω is a nonnegative linear combination of simple roots. A weight ω ∈ X is called dominant if it is
a non-negative linear combination of the fundamental weights. The Weyl group W of G,
generated by the reflections along the αi , acts on X. Under this action each W -orbit on X
contains a unique dominant weight.
From now on let L be a finite dimensional G-module
over F¯ p . Considering this as T L
module there is a direct sum decomposition L = ω∈X Lω into weight spaces Lω such that
t ∈ T acts by multiplication with ω(t) on Lω . The set of ω ∈ X with Lω 6= {0} is called the
set of weights of L. The set of weights of L is a union of W -orbits and for ω ∈ X, x ∈ W we
have dim(Lω ) = dim(Lωx ).
The following basic results characterize the irreducible representations L via their set of
weights.
Theorem 2.1 (Chevalley). Let G and L be as above.
(a) If L is irreducible then the set of weights of L contains a (unique) element λ such
that for all weights ω of L we have ω 6 λ. This λ is called the highest weight of L, it is
dominant and we have dim(Lλ ) = 1.
(b) An irreducible G-module L is determined up to isomorphism by its highest weight.
(c) For each dominant weight λ ∈ X there is an irreducible G-module L(λ) with highest
weight λ.
A dominant weight λ = a1 ω1 + · · · + al ωl ∈ X is called p-restricted if 0 6 ai 6 p − 1 for
1 6 i 6 l.
The following result of Steinberg shows how all highest weight modules L(λ) of G can
be constructed out of those with p-restricted highest weights.
Theorem 2.2 (Steinberg’s tensor product theorem). Let F0 be the Frobenius automorphism of F¯ p , raising elements to their p-th power. Twisting the G-action on a G-module
2
Small degree representations in defining characteristic
L with F0i , i ∈ Z>0 , we get another G-module which we denote by L(i) . If λ0 , . . . , λn are
p-restricted weights then
L(λ0 + pλ1 + · · · + pn λn ) ∼
= L(λ0 ) ⊗ L(λ1 )(1) ⊗ · · · ⊗ L(λn )(n) .
Finally we need to recall the relation between the irreducible modules of the algebraic
group G and those of the finite group G(q). This is nicely described by Steinberg in [19,
13.3, 11.6].
Theorem 2.3 (Steinberg). Let G and G(q) be as in the introduction. We define a subset Λ
of dominant weights. If G(q) is not a Suzuki or Ree group (i.e., not of type 2B2 , 2G2 or 2F4 )
then Λ = {a1 ω1 + · · · + al ωl | 0 6 ai 6 q − 1 for 1 6 i 6 l}. In the case of Suzuki and Ree
√
groups we define Λ = {a1 ω1 + · · · + al ωl | 0 6 ai 6 q/ p − 1 if αi is a long root, 0 6 ai 6
√
q p − 1 if αi is a short root} (note that q is the square root of an odd power of p = 2, 3 or
2, respectively, in these cases).
Then the restrictions of the G-modules L(λ) with λ ∈ Λ to G(q) form a set of pairwise
inequivalent representatives of all equivalence classes of irreducible F¯ p G(q)-modules.
These results show that the dimensions of the irreducible representations of the groups
G(q) over F¯ p are easy to obtain if we know the dimensions of the representations L(λ) of
the algebraic groups G for p-restricted weights λ.
3. Computation of weight multiplicities
In this section we sketch how we compute the degree of the representation L(λ) for
given root datum of G, highest weight λ and prime p. For almost all p it is the same as for
the algebraic group over the complex numbers with same root datum, respectively its Lie
algebra. In these cases the degree can be computed by a formula of Weyl, see [8, 24.3].
In the other cases no formula is known. But in principle there is an algorithm to compute
the degree. This is described in [8, Exercise 2 of 26.4] and goes back to Burgoyne [2]. This
was also used in [6] to handle some cases in exceptional groups. The idea is to construct a
so-called Weyl module V (λ)Z generically over the integers. By base change this leads to a
module V (λ) for any G with the given root datum over any ring, which has λ as a highest
weight. Over C or over F¯ p for almost all p this is irreducible and so isomorphic to L(λ). In
general L(λ) is a quotient of V (λ).
To construct V (λ)Z one considers the universal enveloping algebra U of the complex
Lie algebra corresponding to the given root datum. It contains a Z-lattice UZ , the Kostant
Z-form of U , which is defined via a Chevalley basis of the Lie algebra. Up to equivalence
there is a unique irreducible highest weight representation V (λ) for U with highest weight
λ. Let 0 6= v ∈ V (λ) be a vector of weight λ (this is unique up to scalar). Then we set
V (λ)Z := UZ v = UZ− v.
We fix an ordering β1 , . . . , βN of the set of positive roots. Then UZ− and UZ+ have Z-bases
labeled by sequences a = (a1 , . . . , aN ) of non-negative integers. Applying such a basis element fa of UZ− to a vector of weight ω we get a vector of weight ω−a1 β1 −· · ·−aN βN (and
L
similarly for basis vectors ea of UZ+ ). So, decomposing V (λ)Z = ω∈X (Vω )Z according
to the weight spaces, we can describe generating sets of (Vω )Z by all non-negative linear
combinations of positive roots which are equal to λ − ω.
There is a non-degenerate bilinear form (·, ·) on V (λ)Z which can be described via this
generating system. Different weight spaces are orthogonal with respect to this form. Let
3
Small degree representations in defining characteristic
a, b be two coefficient vectors as above such that the corresponding linear combination of
the positive roots is the same. Then eb fa v = na,b v is again of weight λ and one defines
( fa v, fb v) := na,b ∈ Z.
Since the form is nondegenerate we can compute the rank of a weight lattice (Vω )Z by
computing the rank of the matrix (na,b ) where a and b are running through all non-negative
linear combinations of positive roots for λ − ω. The dimension of the weight space Lω of
L(λ) is the rank of (na,b ) modulo p.
The coefficients na,b can be computed by simplifying the element eb fa with the help
of the commutator relations in U , see [8, 25.]. These involve the structure constants for
a Chevalley basis of the corresponding Lie algebra which can be computed as described
in [3, 4.2].
In principle this allows the determination of dim(L(λ)) for all G, p and λ. But in practice
these computations can become very long, already in small examples. There are technical
problems like the question how to apply commutator relations most efficiently in order to
compute the integers na,b . Different strategies change the number of steps in the calculation considerably. But the main problem is that the generating sets for the weight spaces
as described above are very redundant. Usually the number of non-negative linear combinations of positive roots which yield λ − ω is much larger than the dimension of Vω . By a
careful choice of the ordering of the positive roots one can reduce the linear combinations
to consider, because for many a = (a1 , . . . , aN ) with ω = λ − a1 β1 − · · · − aN βN there is a
1 6 k 6 N such that λ − a1 β1 − · · · − ak βk is not a weight of V (λ) (and hence fa v = 0).
But the main improvement we get by using results of Jantzen and Andersen, see [12,
II.8.19]. The so-called Jantzen sum formula expresses the determinant of the Gram matrix
of the bilinear form (·, ·) on the lattice (Vω )Z in terms of weight multiplicities of various V (µ) with µ 6 λ. The weight multiplicities of these V (µ) can be efficiently computed
by Freudenthal’s formula, see [8, 22.3]. In particular the formula gives exactly the set of
primes p for which the Weyl module V (λ) is not isomorphic to L(λ). In rare cases it happens that a prime p divides such a determinant exactly once - then we know without further
calculations that dim(Lω ) = dim(Vω ) − 1.
Using these determinants we compute only parts of the matrices (na,b ) corresponding
to a subset of its rows and columns until the submatrix has the full rank dim(Vω ) and the
product of its elementary divisors is equal to the known determinant. Then the rank of
(na,b ) modulo p is the same as the rank of this submatrix modulo p.
With this approach we never need submatrices of (na,b ) of much larger dimension than
dim(Vω ). (In [6] the consideration of the matrices (na,b ) was substituted by computing
somewhat smaller matrices using the action of parabolic subalgebras in cases where λ has
a non-trivial stabilizer in W , but these matrices were still big compared to the dimension
of the weight spaces.) Of course, our method is limited to representations where no single
weight space has a dimension of more than a few thousand.
Actually we can also compute by our approach the Jantzen filtrations of the Weyl modules, see [12, II.8], by computing the elementary divisors of the matrices (na,b ) and not just
their rank. One needs a sophisticated algorithm to compute the exact elementary divisors
of such matrices of dimension bigger than 200, say. We developed the algorithm described
in [14] for this purpose.
We have a collection of computer programs for doing the calculations described above;
they are based on the computer algebra system GAP [17] and the package CHEVIE [5].
It is planned to make them available to other users in form of another package. This will
be built on a new version of CHEVIE for GAP - Version 4 which is currently developed.
4
Small degree representations in defining characteristic
While rewriting the current programs we hope to improve their efficiency and to extend
their functionality. We postpone a much more detailed version of this very sketchy section
until this package is ready.
4. Representations of small degree for groups of small rank
From Theorem 2.2 we see how to construct any highest weight representation L(λ) of G
from those with p-restricted weights by twisting with field automorphisms and tensoring.
Assume now we are given the type of an irreducible root system and a number M ∈ N.
We consider the groups G over F¯ p with this root system for all p at once. We want to find for
all primes p all p-restricted dominant weights λ such that the highest weight representation
L(λ) of the group G over F¯ p has degree smaller or equal M.
Our main tool to restrict this question to a finite number of λ to consider is the following
result by Premet, see [16]. Recall from Section 2 that the set of weights of L(λ) is a union
of W -orbits and that each W -orbit contains a unique representative which is dominant.
Theorem 4.1 (Premet). If the root system of G has different root lengths we assume that
p 6= 2 and if G is of type G2 we also assume p 6= 3. Let λ be a p-restricted dominant weight.
Then the set of weights of L(λ) is the union of the W -orbits of dominant weights ω with
ω 6 λ.
We note that in the exceptional cases with different root lengths and p = 2, respectively
p = 3, the statement of the theorem does actually not hold.
The following remark shows how to compute explicitly the number of weights of L(λ)
as given by Premet’s theorem.
Remark 4.2. (a) Let ω = a1 ω1 + · · · + al ωl be a dominant weight. Then the stabilizer of
ω in the Weyl group W is the parabolic subgroup generated by the reflections along the
simple roots αi for which ai = 0.
(b) For given dominant weight λ one finds the set Ω = {ω| ω dominant and ω 6 λ}
efficiently as follows: Initialize Ω ← {λ}. For each ω ∈ Ω and for each positive root α
compute ω − α. If this is also dominant add it to the set Ω.
Proof. For (a) see [8, 10.3B] and for (b) see [15, Prop.1].
Here is an algorithm for finding a set of candidate highest weights for representations
of G of rank at most a given bound. It shows in particular that for any given bound there is
only a finite number of highest weights which must be considered.
Algorithm 4.3. Input: An irreducible root system and an M ∈ N.
Output: A set A of dominant weights which contains all λ such that there is a prime p
for which λ is p-restricted and a group G over F¯ p corresponding to the given root system
with highest weight representation L(λ) of degree at most M.
(1) For a dominant weight λ we define m(λ) as the number of weights in all W -orbits
of dominant weights ω with ω 6 λ. The numbers m(λ) can be computed using 4.2.
(2) We initialize the set A with all 2-restricted weights λ with m(λ) 6 M.
(3) We choose a linear function γ : X → Z which takes positive values on the simple
roots αi , 1 6 i 6 l (and hence on the fundamental weights).
Then we start to enumerate recursively for n = 0, 1, . . . all dominant weights λ with
γ(λ) = n. (Clearly, for fixed n there is a finite number of such weights.) For each of these
λ we compute m(λ) and if it is 6 M we put λ into A.
5
Small degree representations in defining characteristic
We proceed until we find an interval I = [a, a+max(γ(αi ) | 1 6 i 6 l)] such that a > γ(λ)
for all λ ∈ A and such that for all dominant λ with γ(λ) ∈ I we have m(λ) > M.
(4) If the given root system has roots of different length we add all 2-restricted weights
to A. If there is a component of type G2 we also add all 3-restricted weights.
(5) We return the set A.
Proof. We have to show that we will eventually find an interval I, as described in
step (3), and that all dominant weights which were not considered during the algorithm
lead to modules of dimension > M.
In the cases of root systems with different root lengths and p = 2 or p = 3 for which
Premet’s theorem 4.1 does not give a statement we have put in step (4) all 2-, respectively
3-restricted weights into A. So, from now on we only consider weights and primes for
which Premet’s theorem is applicable. This theorem says that m(λ), defined in (1), is a
lower bound for dim(L(λ)) for all primes p such that λ is p-restricted.
Now we show that in steps (2) and (3) we construct the set A as the set of dominant
weights λ with m(λ) 6 M.
First note that for ω 6 λ dominant we have m(ω) 6 m(λ).
Furthermore let λ = a1 ω1 + · · · + al ωl be a dominant weight which is not 2-restricted,
i.e., one ai > 2. Then λ − αi is also a dominant weight. This follows from the fact that
the coefficients of the simple root αi expressed as linear combination of the fundamental weights is given by the i-th column of the Cartan matrix (which has entries 2 on the
diagonal and non-positive entries elsewhere).
So, for a weight λ with one of the coefficients ai > 2k, k ∈ N, there are at least k smaller
dominant weights and so m(λ) > k. This shows that we are constructing a finite set and
that we will find the interval I in step (3).
Now we show by induction on the value n of γ, starting from the lower bound n = a of
the interval I, that for a dominant weight λ with γ(λ) = n we have m(λ) > M.
This is clear for n ∈ I by construction of the interval I in step (3).
So, assume that γ(λ) > a + max(γ(αi ) | 1 6 i 6 l). If λ is 2-restricted we have found
in step (2) that m(λ) > M (it is not in A by construction of I). Otherwise one coefficient
of lambda, say ai , is > 2. Then λ − αi is dominant and we have a 6 γ(λ − αi ) < γ(λ). By
induction hypothesis we get M < m(λ − αi ) < m(λ).
In Table 1 we fix some bound M for each irreducible root system of rank at most 11,
respectively 17 in case Al .
Table 1: Chosen degree bounds for groups of small rank
Type
M
Type
M
Type
M
Type
M
Type
M
Type
M
A2
400
A12
3000
B2
300
G2
500
A3
500
A13
3000
B3
700
C3
1000
A4
1000
A14
3000
B4
1000
C4
2000
D4
2000
F4
12000
A5
2500
A15
3000
B5
2000
C5
2500
D5
3000
A6
2800
A16
3000
B6
4000
C6
4000
D6
4000
E6
50000
6
A7
3000
A17
3000
B7
5000
C7
6000
D7
5000
E7
100000
A8
4000
A9
6000
A10
10000
A11
12000
B8
7000
C8
10000
D8
10000
E8
100000
B9
8000
C9
10000
D9
15000
B10
10000
C10
10000
D10
18000
B11
12000
C11
12000
D11
20000
Small degree representations in defining characteristic
For each irreducible root system listed in Table 1 we used the given M as input for
Algorithm 4.3. For each weight λ in the output set A and for all primes p for which λ
is p-restricted we computed the exact weight multiplicities of L(λ) using the techniques
described in Section 3.
We remark that the sets A from Algorithm 4.3 are much larger than the list of weights
which actually lead to representations of dimension 6 M, since we used a very rough lower
bound for these dimensions. But for weights of representations of much larger degree we
usually find quite fast some weight multiplicities > 1 which gives better lower bounds for
the dimension. We stop the computation of weight multiplicities when this better estimate
becomes larger than M. In this way almost all of the computation time for our result was
actually spent on those weights which appear in our result tables.
The bounds M were chosen such that the results presented below could be computed
interactively with our programs within about two days using 10 computers in parallel. The
types Al , 12 6 l 6 17, were added upon request of Gunter Malle, who asked to cover in
this note all representations of degree 6 2 dim(G) for a specific application.
Here is our main result.
Theorem 4.4. For any type of root system and number M as given in Table 1 and all primes
p the Tables 6.6 to 6.53 list the p-restricted weights λ such that the representation L(λ) of
the algebraic group G over F¯ p has degree at most M. The exact degree of L(λ) is also given.
Furthermore we describe the centers of the groups G and the action of the fundamental
weights on the center in 6.2. This allows to determine the kernels of the representations
L(λ). The Frobenius-Schur indicators in case p 6= 2 are given by 6.3.
As mentioned above we have actually computed the exact weight multiplicities for the
representations appearing in the tables of Section 6. It would take too much space to print
this in detail but the results are available upon request from the author.
Our table for type F4 includes a description of the representations of the Ree group
2 F (2). The related finite simple group is the commutator subgroup which is of index 2.
4
The restrictions of the irreducible representations of 2 F4 (2) in characteristic 2 remain irreducible, except that with highest weight λ = (1100) which splits into two representations
of degree 2048, see the Modular Atlas [10].
The types considered above do not include A1 , i.e., G = SL2 (F¯ p ). The reason is that
this case is easy to describe systematically. This seems to be well known but also follows
immediately from 4.1 since all weight multiplicities are at most 1 in this case, see [8, 7.2].
Remark 4.5. If G is of type A1 then the representation L(kω1 ) with 0 6 k 6 p − 1 has
degree k + 1. For odd p the center of G is non-trivial and of order 2. It is contained in the
kernel of L(kω1 ) if and only if k is even.
5. Representations of small degree for groups of large rank
In this section we consider classical groups of large rank. We prove the following result.
Theorem 5.1. Let G be of classical type and l be the rank of G. Set M = l 3 /8 if G is of
type Al and M = l 3 otherwise. If l > 11 then all p-restricted weights λ such that the highest
weight representation L(λ) of G has dimension at most M are given in Table 2. The table
also includes the dimensions of these modules. (Note that the case of type Bl and p = 2 is
included in case Cl and p = 2.)
The fundamental weights are labeled as explained in 6.1.
7
Small degree representations in defining characteristic
Table 2: Small degrees for classical groups of large rank
type
Al
Bl
Cl
Dl
λ
0
ω1 , ωl
ω2 , ωl−1
2ω1 , 2ωl
ω1 + ωl
ω1 + ωl
0
ωl
ωl−1
2ωl
2ωl
0
ωl
ωl−1
ωl−1
2ωl
0
ωl
ωl−1
ωl−1
2ωl
2ωl
p
all
all
all
all
p | l +1
p - l +1
all
6= 2
6= 2
p | 2l + 1
p - 2l + 1
all
all
p|l
p-l
all
all
all
2
6= 2
p|l
p-l
degree
1
l +1
l(l + 1)/2
(l + 1)(l + 2)/2
l 2 + 2l − 1
l 2 + 2l
1
2l + 1
2l 2 + l
2l 2 + 3l − 1
2l 2 + 3l
1
2l
2l 2 − l − 2
2l 2 − l − 1
2l 2 + l
1
2l
2l 2 − l − gcd(2, l)
2l 2 − l
2l 2 + l − 2
2l 2 + l − 1
Note that the corresponding result for G of rank l 6 11 is included in Theorem 4.4.
Proof. We first prove that the weights which don’t appear in our table correspond to
representations of degree larger than M.
(Case Bl , Cl ) Let λ = a1 ω1 + · · · al ωl be a dominant weight. If some ai 6= 0 then the
W -stabilizer of λ is contained in a reflection subgroup of type Bi−1 × Al−i , see 4.2. Hence,
the W -orbit of λ and so dim(L(λ)) is at least
l
bi := (2l l!)/(2i−1 (i − 1)! · (l − i + 1)!) = 2l−i+1
.
i−1
For l > 12 and 1 6 i 6 l − 2 we have bi > l 3 . If al−1 6= 0 and al 6= 0 then the stabilizer
of λ is contained in a reflection group of type Bl−2 , whose index is > l 3 for all l > 6.
This shows that only weights with at most one non-zero coefficient, either al−1 or al ,
can appear in our list. If al−1 > 2 then, as explained in the proof of 4.3, λ − αl−1 6 λ is
also a dominant weight. But this has coefficient 1 at ωl−2 . Using the estimate as above
for this smaller weight and Premet’s theorem 4.1 (note that the λ considered now is not
2-restricted) we see again that dim(L(λ)) > l 3 . A similar argument shows that al < 3 for
weights in our list.
8
Small degree representations in defining characteristic
So, the only weights which could (and actually do) lead to degrees 6 l 3 are 0, ωl , ωl−1
and 2ωl .
(Case Dl ) Here we find the relevant λ with very similar arguments as in case Bl and Cl .
(Case Al ) Because of the symmetry of the Dynkin diagram the weights a1 ω1 +· · ·+al ωl
and al ω1 + al−1 ω2 + · · · + a1 ωl must describe representations of equal degree (in fact they
are dual to each other). We can again use very similar arguments as above to find the
relevant weights for our list. (Here we rule out 3ω1 and ω1 + ω2 by observing 3ω1 − 2α1 −
α2 = ω3 = ω1 + ω2 − α1 − α2 .)
It remains to determine the exact degrees of the L(λ) in our table. Probably they are
known to the experts but we could not find references for all cases. Type Al and L(ωl ) and
L(ωl−1 ) for the other types are contained in [13, 5.4.11] and [8, 25.5, Ex. 8].
We include a proof for the degrees of L(2ωl ) and L(ωl−1 ) in types Bl , Cl and Dl following hints of K. Magaard and G. Hiß.
The idea is to use explicit modules. For all types we know the G-modules V (Xl ) :=
L(ωl ), X ∈ {A, B,C, D}. These are the natural modules of SLl+1 , SO2l+1 , Sp2l or SO2l ,
respectively. We determine the constituents of V (Xl ) ⊗V (Xl ) for X ∈ {B,C, D}. Note that
for these types V (Xl ) is self-dual. Since the case p = 2 is covered by the references above
(2ωl is not 2-restricted), we assume in the rest of the proof that p is odd.
We will use the following general remarks. If V is an indecomposable F¯ p G-module
then V ⊗ V ∗ has the trivial module as direct summand if and only if p - dim(V ) (here V ∗
denotes the dual module). In that case there is exactly one trivial direct summand, see [1,
3.1.9] for a proof. If v1 , . . . , vn is a basis of V then V ⊗V has two submodules S2 (V ) with
basis {vi ⊗ v j + v j ⊗ vi | 1 6 i < j 6 n} ∪ {vi ⊗ vi | 1 6 i 6 n} and Λ2 (V ) with basis
{vi ⊗ v j − v j ⊗ vi | 1 6 i < j 6 n}. Since p 6= 2 we have V ⊗V = S2 (V ) ⊕ Λ2 (V ).
(Case Cl ) Let {vi , v0i | 1 6 i 6 l} be a basis of the natural symplectic module V (Cl ) such
that for the symplectic form (·, ·) we have (vi , v0i ) = 1 = −(v0i , vi ), 1 6 i 6 l and (vi , v j ) =
0 = (vi , v0j ) for all i 6= j. Then the vector ∑li=1 (vi ⊗ v0i − v0i ⊗ vi ) ∈ Λ2 (V ) is invariant under
the symplectic group. If p - 2l this vector must span the unique trivial direct summand
mentioned above.
The group G contains a subgroup isomorphic to SLl (F¯ p ). An element acts on the subspace of V (Cl ) spanned by v1 , . . . , vl and by its inverse on the subspace spanned by v01 , . . . v0l .
Hence V (Cl ) restricted to this subgroup is isomorphic to V (Al−1 ) ⊕V (Al−1 )∗ . Restricting
the representation S2 (V (Cl )) to this subgroup we find the decomposition S2 (V (Al−1 )) ⊕
S2 (V (Al−1 )∗ ) ⊕ (V (Al−1 ) ⊗ V (Al−1 )∗ ), see [4, I,12.]. Using the known degrees for case
Al−1 , we see that this contains at most 2 trivial constituents. The similar restriction to a
subgroup of type Cl−1 shows by induction that one irreducible constituent of S2 (V (Cl ))
has degree at least 2(l − 1)2 + l − 1. Comparing degrees we see that the three non-trivial
constituents in the restriction to SLl must lie in a single constituent of S2 (V (Cl )). To summarize, we have at most one non-trivial and two trivial constituents. If there are trivial
constituents then one must be in the socle, i.e., there must be an invariant vector. This can
only be the one we see in the (V (Al−1 ) ⊗V (Al−1 )∗ )-summand of the restriction to SLl . We
can write down such a vector and check that it is not invariant under the whole group G;
apply an element which does not leave the space spanned by v1 , . . . , vl invariant. We have
proved that S2 (V (Cl )) is irreducible.
We can argue very similarly for Λ2 (V (Cl )). If p - l we find that it is a direct sum of an
irreducible and the trivial module found above. If p | l we find that there is one non-trivial
constituent, exactly one trivial constituent in the socle and at most two trivial constituents.
9
Small degree representations in defining characteristic
Since in this case the trivial constituent in the socle is not a direct summand there must be
a second trivial constituent in the head, by duality.
If V (Cl ) has a highest weight vector v then v ⊗ v is contained in S2 (V (Cl )) and has
weight 2ωl . Hence S2 (V (Cl )) ∼
= L(2ωl ). The other constituents of the tensor product must
correspond to dominant weights smaller than 2ωl , there are only two of them, 2ωl − αl =
ωl−1 and 0. We get that the non-trivial constituent of Λ2 (V (Cl )) is isomorphic to L(ωl−1 ).
(Case Dl ) This can be handled by almost exactly the same arguments as the case Cl .
Here the trivial submodule is contained in S2 (V (Dl )).
(Case Bl ) In this case we consider the restrictions of S2 (V (Bl )) to subgroups Hr of type
Br + Dl−r , 0 6 r < l. This leads to decompositions S2 (V (Br )) ⊕ S2 (V (Dl−r )) ⊕ (V (Dl−r ) ⊗
V (Br )). Comparing this decomposition for r = 0 and r = l/2 or r = (l − 1)/2, respectively,
and using the results for type Dl−r and induction we see as in type Cl that S2 (V (Bl )) has
only one non-trivial and maybe a few trivial constituents. There is an r such that p - (l − r)
and p - (2r + 1). The decomposition above for this r shows that there are at most two trivial
constituents. Furthermore, as in type Dl we find a trivial submodule. Either this is a direct
summand (if p - (2l + 1)) then it is the only trivial constituent or otherwise there is a second
constituent in the head of S2 (V (Bl )). The argument for Λ2 (V (Bl )) is again very similar.
This finishes the proof.
It would be interesting if the degrees in our list could be determined more systematically
in the framework of highest weight modules. Then one could work out systematically
generalizations of Theorem 5.1 where the bound M is substituted by any fixed polynomial
in l.
The following Corollary completes the list in [7] which gives all representations of G(q)
in non-defining characteristic of degree at most 250.
Corollary 5.2. For any simple G over F¯ p we find all p-restricted weights λ of G such that
L(λ) has degree 6 250 in the tables given in Theorems 4.4 and 5.1. Taking Theorem 2.2
into account this determines all L(λ) with degree 6 250.
6. Appendix: Tables for groups of small rank
In this section we give the detailed lists for Theorem 4.4. We start by introducing some
notation and describe the centers of the groups G and the action of the fundamental weights
on the center.
6.1. Ordering of fundamental weights
For the irreducible types of root systems we choose the following ordering for the simple
roots αi , the corresponding coroots α∨i and the fundamental weights ωi , 1 6 i 6 l. We show
the Dynkin diagrams with the node of αi labeled by i. (This is the labeling we use in all
databases of the CHEVIE [5] project.)
10
Small degree representations in defining characteristic
Al
Dl
s
1
s
2
s1
@ 3
@s
3
s
p p p
l
s
p p p
l
4
s
Bl
s
Cl
1
s < 2s
3
s
p p p
l
s > 2s
3
s
p p p
l
1
s
3
s
4
s
5
s
6
s
5
s
6
s
8
1
s
s
s2
G2
s > 2s
1
s
s > 3s
1
F4
2
s
4
E6
s
s2
E7
s
1
s
3
s
4
s
5
s
6
s
7
E8
s2
s
1
s
3
4
s
7
s
s2
6.2. Action of fundamental weights on the center of G
We give for each irreducible type of root system the values of ωi (z) for z in the center
Z(G) of G.
To compute this we use that Z(G) is contained in a maximal torus T of G. Such a T is
isomorphic to (∑li=1 Zα∨i ) ⊗Z F¯ ×
p and Z(G) consists of those z ∈ T with αi (z) = 1 for all
1 6 i 6 l. So, we have to solve a system of equations given by the Cartan matrix of the root
system.
We denote ζm ∈ F¯ ×
p an element whose multiplicative order is m. For n ∈ N we write n p0
for the largest divisor of n which is prime to p. For elements z ∈ Z(G) we write ω(z) :=
(ω1 (z), . . . , ωl (z)).
(Al ) Z(G) is cyclic of order m = (l + 1) p0 . It contains a generator z such that ω(z) =
(ζm , ζ2m , . . . ζlm ).
(Bl ) Z(G) is cyclic of order m = gcd(2, p + 1). For the generator z we have ω(z) =
(ζm , 1, . . . , 1).
(Cl ) Z(G) is cyclic of order m = gcd(2, p + 1). For the generator z we have ω(z) =
(ζlm , ζl−1
m , . . . , 1, ζm )
(Dl , l odd) Z(G) is cyclic of order m = 4 for odd p and m = 1 for p = 2. There is a
generator z such that ω(z) = (ζm , ζ3m , ζ2m , 1, ζ2m , 1, ζ2m , . . . , 1, ζ2m ).
(Dl , l even) Z(G) is elementary abelian of order m2 with m = 2 if p is odd and m = 1 if
p = 2. There are generators z1 and z2 such that ω(z1 ) = (ζm , 1, 1, ζm , 1, ζm , . . . , 1, ζm )
and ω(z2 ) = (1, ζm , 1, ζm , 1, ζm , . . . , 1, ζm ). Here z = z1 z2 is the element such that
G/hzi ∼
= SO2l (F¯ p ).
(E6 ) Z(G) is cyclic of order m = 3 if p 6= 3 and m = 1 if p = 3. There is a generator z such
that ω(z) = (ζm , 1, ζ2m , 1, ζm , ζ2m ).
(E7 ) Z(G) is cyclic of order m = gcd(2, p + 1). For the generator z we have ω(z) =
(1, ζm , 1, 1, ζm , 1, ζm ).
(G2 , F4 , E8 ) Z(G) is trivial.
6.3. Frobenius-Schur indicators
If p is odd we can determine the Frobenius-Schur indicators of the representations in
our lists using a result of Steinberg, see [18, Lemmas 78 and 79].
11
Small degree representations in defining characteristic
If G is of type Al or of type Dl with odd l or of type E6 then the representations
L(∑li=1 ai ωi ) and L(∑li=1 aτ(i) ωi ), where the permutation τ is given by the automorphism of
order two of the Dynkin diagram, are dual to each other. For other G all L(λ) are self-dual.
If L(λ) is self-dual (and recall that p is odd) then its Frobenius-Schur indicator is ‘+’ if
Z(G) has no element of order 2. Otherwise it is the sign of λ(z) where z is the only element
of order 2 in Z(G), respectively z = z1 z2 in case Dl with even l. This can be computed
by 6.2.
6.4. An example
As an example how to read the data in this Appendix, let us determine all irreducible representations in defining characteristic for groups of type B3 (q) ∼
= Spin7 (q), q = p f which
have degree 448:
We apply Theorems 2.2 and 2.3.
We first need to compute all factorizations of 448 into factors > 1 which appear as degrees in 6.23. Although many divisors of 448 appear as degree the only such factorizations
are 448 = 7 · 64 = 7 · 8 · 8.
Now we have a closer look at 6.23.
If p = 2 then there is no irreducible representation of degree 7. And the listed representations of degree 448 do not correspond to 2-restricted weights. So, in characteristic 2
there are no irreducible representations of this degree.
For p 6= 2 we have representations with p-restricted weights of dimension 7 and 8. If
q > p3 there are irreducible representations Li, j,k of degree 448 with highest weight of form
pi ω3 + p j ω1 + pk ω1 , where 0 6 i, j, k 6 f − 1 and i, j, k are pairwise different.
For p = 3 there is no irreducible representation of degree 64 and none with p-restricted
weight of degree 448. So, there are no further irreducible representations of degree 448 in
this case.
If p = 5 there is L(ω3 ) of degree 7 and L(ω1 + ω2 ) of degree 64. If q = 5 f we find
for any 0 6 j, k 6 f − 1, j 6= k, the irreducible representation L0j,k := L(5 j (ω1 + ω2 ) +
5k ω3 ) restricted to B3 (q) of degree 448. Furthermore we see in 6.23 that L(ω1 + 3ω3 ) and
L(2(ω1 + ω3 )) have degree 448.
For larger p there is no representation of degree 64 in our list. We only find L(ω1 + 3ω3 )
if p 6= 11 and in case p = 7 also L(3ω1 + ω2 ).
For all representations found above the prime p is odd and so the center of G and of
G(q) is of order 2. Using 6.2 we see that for the non-trivial element z in the center we have
ω1 (z) = −1 and ωi (z) = 1 for i = 2, 3. Applying this to the weights listed above we find
that exactly L(2(ω1 + ω3 )) in case p = 5 and Li, j,k for p 6= 2 are not faithful. From 6.3 we
see that these are also the representations with Frobenius-Schur indicator ‘+’. The others
have indicator ‘-’.
6.5. Reading the tables
In the tables below we denote a weight a1 ω1 + · · · + al ωl by (a1 , . . . , al ). When all ai
can be written with a single digit we also suppress the commas.
In type Al the representations L(a1 ω1 + · · · + al ωl ) and L(al ω1 + al−1 ω2 + · · · + a1 ωl )
are dual to each other, in particular they have the same dimension. In 6.6 to 6.21 we save
some space by only including one of such a pair of representations.
12
Small degree representations in defining characteristic
6.6. Case A2 , M = 400
(Recall the remark in 6.5.)
deg
1
3
6
7
8
10
15
15
18
19
21
24
27
28
33
35
36
36
37
39
42
45
48
55
60
63
63
64
66
71
75
75
78
80
81
82
87
90
90
91
91
99
102
105
105
111
114
117
118
120
120
λ
(00)
(01)
(02)
(11)
(11)
(03)
(04)
(12)
(13)
(22)
(05)
(13)
(22)
(06)
(15)
(14)
(07)
(24)
(33)
(23)
(23)
(08)
(15)
(09)
(24)
(16)
(33)
(33)
(0,10)
(25)
(19)
(34)
(0,11)
(17)
(25)
(28)
(37)
(34)
(46)
(0,12)
(55)
(18)
(1,11)
(0,13)
(26)
(2,10)
(35)
(44)
(39)
(0,14)
(19)
p
all
all
all
3
6=3
all
all
all
5
5
all
6=5
6=5
all
7
all
all
7
7
5
6=5
all
6=7
all
6=7
all
5
6=5,7
all
7
11
7
all
all
6=7
11
11
6=7
11
all
11
all
13
all
all
13
7
7
13
all
6=11
deg
120
123
125
126
127
132
136
143
153
154
159
162
162
165
168
168
171
171
179
181
183
190
192
192
195
195
201
207
208
210
210
213
215
215
216
216
217
222
224
231
231
231
234
235
243
246
251
252
253
255
255
λ
(35)
(48)
(44)
(57)
(66)
(27)
(0,15)
(1,10)
(0,16)
(36)
(29)
(28)
(45)
(45)
(1,11)
(1,15)
(0,17)
(38)
(47)
(2,14)
(56)
(0,18)
(37)
(3,13)
(1,12)
(29)
(4,12)
(1,17)
(5,11)
(0,19)
(46)
(6,10)
(2,11)
(55)
(55)
(79)
(88)
(2,16)
(1,13)
(0,20)
(2,10)
(3,10)
(38)
(3,15)
(49)
(4,14)
(58)
(39)
(0,21)
(1,14)
(5,13)
13
p
6=7
13
6=7
13
13
all
all
all
all
all
11
6=11
7
6=7
6=13
17
all
11
11
17
11
all
6=11
17
all
6=11
17
19
17
all
6=11
17
13
7
6=7,11
17
17
19
all
all
6=13
13
6=11
19
13
19
13
11
all
all
19
deg
255
260
262
267
267
270
270
271
273
276
276
279
280
288
297
300
312
315
316
323
325
330
333
336
339
343
348
351
351
354
357
360
360
361
370
372
375
375
375
378
378
381
384
384
387
388
393
395
396
397
399
λ
(67)
(47)
(6,12)
(48)
(7,11)
(2,11)
(8,10)
(99)
(56)
(0,22)
(57)
(66)
(39)
(1,15)
(1,21)
(0,23)
(2,12)
(48)
(2,20)
(1,16)
(0,24)
(3,10)
(3,19)
(57)
(3,11)
(66)
(4,18)
(0,25)
(2,15)
(49)
(2,13)
(1,17)
(4,10)
(5,17)
(58)
(6,16)
(3,14)
(49)
(59)
(0,26)
(67)
(7,15)
(3,11)
(68)
(77)
(8,14)
(9,13)
(4,13)
(10,12)
(11,11)
(1,18)
p
13
6=11
19
11
19
6=13
19
19
6=11
all
11
11
6=11,13
6=17
23
all
all
6=11,13
23
all
all
6=13
23
6=11,13
13
6=11,13
23
all
17
11
all
6=19
13
23
11
23
17
6=11,13
13
all
11
23
6=13
13
13
23
23
17
23
23
all
Small degree representations in defining characteristic
6.7. Case A3 , M = 500
(Recall the remark in 6.5.)
deg
1
4
6
10
14
15
16
19
20
20
20
32
35
36
44
45
50
52
56
58
60
64
68
69
70
80
83
84
84
84
85
100
105
116
116
λ
(000)
(001)
(010)
(002)
(101)
(101)
(011)
(020)
(003)
(011)
(020)
(102)
(004)
(102)
(111)
(012)
(030)
(013)
(005)
(111)
(021)
(111)
(022)
(202)
(103)
(031)
(202)
(006)
(013)
(202)
(040)
(104)
(040)
(015)
(112)
p
all
all
all
all
2
6=2
3
3
all
6=3
6=3
5
all
6=5
3
all
all
5
all
5
all
6=3,5
5
3
all
5
5
all
6=5
6=3,5
5
7
6=5
7
3
deg
120
120
124
126
140
140
140
149
156
160
165
173
175
180
184
189
192
196
206
211
216
220
220
224
224
231
235
236
256
260
260
270
280
280
280
λ
(007)
(104)
(203)
(022)
(014)
(031)
(112)
(024)
(121)
(203)
(008)
(113)
(121)
(033)
(131)
(105)
(023)
(050)
(042)
(113)
(015)
(009)
(122)
(023)
(051)
(060)
(032)
(212)
(113)
(041)
(204)
(204)
(032)
(041)
(106)
14
p
all
6=7
7
6=5
all
6=5
6=3
7
3
6=7
all
5
6=3
7
5
all
5
all
7
7
6=7
all
5
6=5
7
7
5
7
6=5,7
5
7
6=7
6=5
6=5
all
deg
285
286
294
299
300
300
300
315
334
336
360
360
360
364
380
380
384
396
416
420
420
420
420
439
440
455
460
476
480
480
484
496
500
λ
(303)
(0,0,10)
(212)
(222)
(122)
(212)
(303)
(016)
(131)
(060)
(019)
(024)
(122)
(0,0,11)
(115)
(213)
(131)
(107)
(025)
(108)
(114)
(205)
(313)
(028)
(017)
(0,0,12)
(033)
(132)
(033)
(124)
(151)
(304)
(304)
p
7
all
3
5
7
6=3,7
6=7
all
7
6=7
11
6=7
6=5,7
all
7
5
6=5,7
all
7
11
all
all
5
11
all
all
5
5
6=5,7
7
7
7
6=7
Small degree representations in defining characteristic
6.8. Case A4 , M = 1000
(Recall the remark in 6.5.)
deg
1
5
10
15
23
24
30
35
40
40
45
45
50
51
65
70
70
74
75
103
105
121
126
126
135
145
160
160
165
170
174
175
175
175
176
λ
(0000)
(0001)
(0010)
(0002)
(1001)
(1001)
(0011)
(0003)
(0011)
(0101)
(0020)
(0101)
(0020)
(0110)
(1002)
(0004)
(1002)
(0110)
(0110)
(0102)
(0012)
(0013)
(0005)
(0102)
(1011)
(1003)
(1003)
(1011)
(0111)
(0201)
(0021)
(0021)
(0030)
(1011)
(2002)
p
all
all
all
all
5
6=5
3
all
6=3
2
3
6=2
6=3
3
3
all
6=3
2
6=2,3
5
all
5
all
6=5
3
7
6=7
2
3
5
3
6=3
all
6=2,3
7
deg
185
195
199
200
210
210
224
235
255
280
280
305
315
315
320
325
330
365
375
381
390
395
410
410
420
420
435
445
450
450
470
476
480
490
495
λ
(0022)
(0201)
(2002)
(2002)
(0006)
(0201)
(0013)
(0111)
(0031)
(0103)
(0111)
(0120)
(0120)
(1004)
(0040)
(0015)
(0007)
(0130)
(1102)
(0220)
(0202)
(0104)
(1012)
(1102)
(0014)
(0022)
(1102)
(2003)
(1012)
(2003)
(0024)
(1111)
(1102)
(0040)
(0008)
15
p
5
3
3
6=3,7
all
6=3,5
6=5
5
5
all
6=3,5
5
6=5
all
5
7
all
5
3
5
3
7
7
5
all
6=5
7
7
6=7
6=7
7
3
6=3,5,7
6=5
all
deg
510
535
540
545
560
560
560
615
640
670
670
683
700
700
715
720
720
765
794
826
835
840
855
875
895
924
945
945
949
955
980
999
1000
λ
(0112)
(1013)
(0104)
(0202)
(0031)
(0202)
(1005)
(1021)
(0033)
(0023)
(1021)
(1022)
(0112)
(0301)
(0009)
(0015)
(1021)
(0203)
(1111)
(0042)
(0211)
(0023)
(0121)
(2004)
(0113)
(1006)
(0105)
(1013)
(1111)
(0032)
(0130)
(3003)
(3003)
p
3
5
6=7
5
6=5
6=3,5
all
5
7
5
7
5
6=3
all
all
6=7
6=5,7
7
5
7
3
6=5
3
all
5
all
all
6=5
7
5
6=5
7
6=7
Small degree representations in defining characteristic
6.9. Case A5 , M = 2500
(Recall the remark in 6.5.)
deg
1
6
15
20
21
34
34
35
50
56
70
78
84
90
90
105
105
114
120
126
126
141
154
175
188
189
204
210
210
246
246
252
258
279
280
315
336
363
369
384
400
404
405
414
420
λ
(00000)
(00001)
(00010)
(00100)
(00002)
(10001)
(10001)
(10001)
(00011)
(00003)
(00011)
(01001)
(01001)
(00020)
(00101)
(00020)
(00101)
(10002)
(10002)
(00004)
(00110)
(00200)
(01010)
(00200)
(01010)
(01010)
(00110)
(00012)
(00110)
(00013)
(01002)
(00005)
(00102)
(10011)
(01002)
(10003)
(00102)
(10011)
(10011)
(10011)
(10101)
(20002)
(20002)
(00021)
(00021)
p
all
all
all
all
all
2
3
6=2,3
3
all
6=3
5
6=5
3
2
6=3
6=2
7
6=7
all
3
3
2
6=3
5
6=2,5
2
all
6=2,3
5
3
all
5
3
6=3
all
6=5
5
7
6=3,5,7
2
7
6=7
3
6=3
deg
426
440
462
474
490
504
520
540
560
606
630
630
666
666
700
708
720
720
720
786
792
804
813
840
840
840
896
924
951
960
960
980
1050
1050
1050
1071
1078
1098
1134
1134
1170
1176
1176
1194
1224
λ
(00022)
(02001)
(00006)
(00111)
(00030)
(00013)
(10101)
(10101)
(02001)
(01003)
(00201)
(01011)
(00031)
(01101)
(10004)
(00111)
(00201)
(01003)
(01011)
(00015)
(00007)
(11002)
(01020)
(00103)
(00201)
(01011)
(00111)
(01101)
(00040)
(00300)
(01110)
(00300)
(00014)
(01101)
(20003)
(01020)
(10012)
(00120)
(00022)
(10012)
(00120)
(00120)
(01020)
(00104)
(10013)
16
p
5
3
all
3
all
6=5
7
6=2,7
6=3
7
5
3
5
3
all
5
3
6=7
2
7
all
3
5
all
6=3,5
6=2,3
6=3,5
2
5
5
3
6=5
all
6=2,3
all
3
5
5
6=5
6=5
3
6=3,5
6=3,5
7
5
deg
1246
1251
1260
1287
1338
1365
1386
1420
1431
1449
1470
1506
1539
1569
1575
1674
1686
1701
1751
1764
1764
1800
1800
1876
1890
1960
1974
1980
1995
2002
2024
2061
2100
2106
2205
2268
2290
2310
2310
2364
2415
2430
2430
λ
(00130)
(00024)
(11002)
(00008)
(10102)
(00210)
(10005)
(01110)
(02002)
(00210)
(00210)
(00220)
(00202)
(02002)
(01004)
(00112)
(00310)
(10102)
(00400)
(00031)
(00040)
(00104)
(02002)
(00033)
(00023)
(01110)
(10021)
(00015)
(20004)
(00009)
(11011)
(01012)
(03001)
(01102)
(10021)
(10006)
(30003)
(10022)
(20004)
(10201)
(00202)
(01012)
(10111)
p
5
7
6=3
all
5
5
all
5
7
3
6=3,5
5
3
3
all
3
5
6=5
5
6=5
6=5
6=7
6=3,7
7
5
6=3,5
3
6=7
5
all
3
7
7
3
6=3
11
5
5
6=5
5
5
6=7
3
Small degree representations in defining characteristic
6.10. Case A6 , M = 2800
(Recall the remark in 6.5.)
deg
1
7
21
28
35
47
48
77
84
112
133
133
140
161
175
189
196
203
210
210
224
266
344
357
378
391
392
393
448
455
462
469
483
λ
(000000)
(000001)
(000010)
(000002)
(000100)
(100001)
(100001)
(000011)
(000003)
(000011)
(010001)
(010001)
(010001)
(000020)
(000101)
(100002)
(000020)
(001001)
(000004)
(000101)
(001001)
(000110)
(010010)
(000200)
(000012)
(010010)
(010010)
(001100)
(001010)
(000013)
(000005)
(000110)
(100011)
p
all
all
all
all
all
7
6=7
3
all
6=3
2
3
6=2,3
3
2
all
6=3
5
all
6=2
6=5
3
5
3
all
3
6=3,5
3
2
5
all
2
3
deg
490
490
493
540
553
560
581
588
687
707
707
735
735
736
756
783
784
840
861
875
882
924
1008
1050
1113
1148
1148
1176
1211
1260
1302
1323
1386
λ
(000110)
(000200)
(010002)
(010002)
(000102)
(100003)
(001010)
(001010)
(200002)
(001002)
(100011)
(100011)
(200002)
(001100)
(000102)
(001100)
(001100)
(001002)
(000021)
(000022)
(000021)
(000006)
(000013)
(020001)
(100101)
(000111)
(020001)
(000030)
(100101)
(020001)
(100004)
(100101)
(100004)
17
p
6=2,3
6=3
7
6=7
5
all
5
6=2,5
3
3
2
6=2,3
6=3
2
6=5
5
6=2,3,5
6=3
3
5
6=3
all
6=5
3
2
3
7
all
5
6=3,7
5
6=2,5
6=5
deg
1428
1520
1575
1701
1709
1716
1771
1827
1863
1876
1907
1953
1967
1967
2016
2100
2106
2155
2156
2198
2310
2331
2352
2387
2394
2400
2400
2415
2450
2611
2646
2646
2800
λ
(010011)
(000031)
(010003)
(011001)
(000015)
(000007)
(000111)
(010011)
(000201)
(010011)
(001003)
(110002)
(002001)
(200003)
(010011)
(000103)
(000201)
(001011)
(200003)
(010101)
(000014)
(100012)
(000111)
(010020)
(001101)
(001003)
(001011)
(000040)
(100012)
(110002)
(000022)
(000201)
(110002)
p
3
5
all
3
7
all
5
5
5
7
7
3
3
5
6=3,5,7
all
3
3
6=5
2
all
3
6=3,5
3
3
6=7
2
5
6=3
7
6=5
6=3,5
6=3,7
Small degree representations in defining characteristic
6.11. Case A7 , M = 3000
(Recall the remark in 6.5.)
deg
1
8
28
36
56
62
63
70
112
120
168
208
216
266
272
280
308
330
336
378
392
392
420
448
504
504
630
657
λ
(0000000)
(0000001)
(0000010)
(0000002)
(0000100)
(1000001)
(1000001)
(0001000)
(0000011)
(0000003)
(0000011)
(0100001)
(0100001)
(0000020)
(1000002)
(1000002)
(0000101)
(0000004)
(0000020)
(0000101)
(0010001)
(0010001)
(0010001)
(0001001)
(0000110)
(0001001)
(0000012)
(0100010)
p
all
all
all
all
all
2
6=2
all
3
all
6=3
7
6=7
3
3
6=3
2
all
6=3
6=2
2
3
6=2,3
5
3
6=5
all
3
deg
658
719
720
784
784
792
860
888
924
945
952
1008
1016
1064
1092
1107
1128
1169
1176
1231
1232
1244
1280
1336
1344
1484
1512
1512
λ
(0100010)
(0100010)
(0100010)
(0000013)
(0000200)
(0000005)
(1000011)
(1000003)
(1000003)
(0100002)
(0000110)
(0000110)
(0001100)
(0000102)
(0001010)
(0002000)
(0010010)
(2000002)
(0000200)
(2000002)
(2000002)
(1000011)
(1000011)
(0010010)
(0010010)
(0001010)
(0000102)
(0001010)
p
2
7
6=2,3,7
5
3
all
3
5
6=5
all
2
6=2,3
3
5
2
3
5
5
6=3
3
6=3,5
7
6=3,7
3
6=3,5
5
6=5
6=2,5
deg
1592
1624
1632
1652
1680
1708
1716
1763
1764
1800
1848
2016
2100
2128
2136
2289
2344
2352
2352
2400
2464
2520
2520
2520
2584
2632
2800
2828
λ
(0010002)
(0000021)
(0010100)
(0000022)
(0000021)
(0001002)
(0000006)
(0002000)
(0002000)
(0010002)
(0000013)
(0200001)
(0001002)
(1000101)
(0001100)
(0010100)
(0001100)
(0001100)
(0010100)
(1000004)
(0000111)
(0000030)
(0200001)
(1000004)
(0100011)
(1000101)
(1000101)
(1001001)
p
7
3
2
5
6=3
3
all
5
6=3,5
6=7
6=5
3
6=3
2
2
5
5
6=2,3,5
6=2,5
11
3
all
6=3
6=11
3
3
6=2,3
5
λ
(00010001)
(00010001)
(00000012)
(00001001)
(00010001)
(01000010)
(01000010)
(01000010)
(00000013)
(00000005)
(10000011)
(10000003)
(10000003)
(01000002)
(01000002)
(00000200)
(00000110)
(20000002)
(00000102)
(00000110)
(20000002)
(20000002)
(10000011)
(10000011)
p
2
3
all
6=5
6=2,3
7
2
6=2,7
5
all
3
11
6=11
3
6=3
3
2
11
5
6=2,3
5
6=5,11
2
5
deg
2079
2304
2352
2385
2394
2520
2691
2700
2772
2844
2907
2922
2970
3003
3060
3139
3168
3318
3402
3414
3465
3654
3744
3780
λ
(10000011)
(00001100)
(00001010)
(00100010)
(00100010)
(00000200)
(00100010)
(00100010)
(00000102)
(00000021)
(00002000)
(00000022)
(00000021)
(00000006)
(00010010)
(00011000)
(00000013)
(00001010)
(00001010)
(02000001)
(00100002)
(00001002)
(00010010)
(00010010)
p
6=2,3,5
3
2
3
2
6=3
7
6=2,3,7
6=5
3
3
5
6=3
all
5
3
6=5
5
6=2,5
3
all
3
3
6=3,5
6.12. Case A8 , M = 4000
(Recall the remark in 6.5.)
deg
1
9
36
45
79
80
84
126
156
165
240
306
315
387
396
414
495
504
540
630
684
720
882
882
λ
(00000000)
(00000001)
(00000010)
(00000002)
(10000001)
(10000001)
(00000100)
(00001000)
(00000011)
(00000003)
(00000011)
(01000001)
(01000001)
(10000002)
(10000002)
(00000020)
(00000004)
(00000101)
(00000020)
(00000101)
(00100001)
(00100001)
(00000110)
(00001001)
p
all
all
all
all
3
6=3
all
all
3
all
6=3
2
6=2
5
6=5
3
all
2
6=3
6=2
7
6=7
3
5
deg
966
966
990
1008
1050
1135
1214
1215
1278
1287
1359
1395
1440
1461
1540
1554
1764
1864
1890
1890
1943
1944
2034
2043
18
Small degree representations in defining characteristic
6.13. Case A9 , M = 6000
(Recall the remark in 6.5.)
deg
1
10
45
55
98
98
99
120
210
210
220
252
330
430
440
530
540
615
715
780
825
990
λ
(000000000)
(000000001)
(000000010)
(000000002)
(100000001)
(100000001)
(100000001)
(000000100)
(000000011)
(000001000)
(000000003)
(000010000)
(000000011)
(010000001)
(010000001)
(100000002)
(100000002)
(000000020)
(000000004)
(000000101)
(000000020)
(000000101)
p
all
all
all
all
2
5
6=2,5
all
3
all
all
all
6=3
3
6=3
11
6=11
3
all
2
6=3
6=2
deg
1110
1155
1452
1485
1596
1826
1848
1860
1924
1925
1980
1990
1992
2002
2100
2100
2145
2278
2310
2376
2826
2850
λ
(001000001)
(001000001)
(000000110)
(000000012)
(000001001)
(010000010)
(000001001)
(000100001)
(010000010)
(010000010)
(000100001)
(100000011)
(000000013)
(000000005)
(000010001)
(000010001)
(100000003)
(010000002)
(000010001)
(010000002)
(200000002)
(000000200)
p
2
6=2
3
all
5
2
6=5
7
3
6=2,3
6=7
3
5
all
2
3
all
5
6=2,3
6=5
3
3
deg
2924
2925
3048
3155
3156
3200
3300
4510
4620
4698
4740
4752
4905
4940
4950
4950
4950
5005
5148
5730
5940
λ
(200000002)
(200000002)
(000000110)
(100000011)
(000000102)
(100000011)
(000000110)
(001000010)
(000001010)
(000000021)
(000001100)
(000000102)
(000000022)
(001000010)
(000000021)
(000000200)
(001000010)
(000000006)
(000000013)
(001000002)
(020000001)
p
11
6=3,11
2
11
5
6=3,11
6=2,3
7
2
3
3
6=5
5
2
6=3
6=3
6=2,7
all
6=5
3
3
6.14. Case A10 , M = 10000
(Recall the remark in 6.5.)
deg
1
11
55
66
119
120
165
275
286
330
440
462
583
583
594
704
715
880
1001
1155
1210
1485
1705
λ
(0000000000)
(0000000001)
(0000000010)
(0000000002)
(1000000001)
(1000000001)
(0000000100)
(0000000011)
(0000000003)
(0000001000)
(0000000011)
(0000010000)
(0100000001)
(0100000001)
(0100000001)
(1000000002)
(1000000002)
(0000000020)
(0000000004)
(0000000101)
(0000000020)
(0000000101)
(0010000001)
p
all
all
all
all
11
6=11
all
3
all
all
6=3
all
2
5
6=2,5
3
6=3
3
all
2
6=3
6=2
3
deg
1760
2145
2277
2706
2784
2903
2904
2959
2992
3003
3014
3080
3168
3300
3391
3465
3510
4115
4158
4158
4234
4235
4422
λ
(0010000001)
(0000000012)
(0000000110)
(0000001001)
(0100000010)
(0100000010)
(0100000010)
(1000000011)
(0000000013)
(0000000005)
(1000000003)
(1000000003)
(0000001001)
(0001000001)
(0100000002)
(0001000001)
(0100000002)
(2000000002)
(0000010001)
(0000010001)
(2000000002)
(2000000002)
(0000100001)
19
p
6=3
all
3
5
3
5
6=3,5
3
5
all
13
6=13
6=5
2
11
6=2
6=11
13
2
3
3
6=3,13
7
deg
4620
4653
4653
4719
4752
4917
4983
5016
5445
7403
7722
7865
7876
7887
8008
8008
8448
8459
8470
9042
9075
9405
9713
λ
(0000010001)
(1000000011)
(1000000011)
(1000000011)
(0000100001)
(0000000200)
(0000000110)
(0000000102)
(0000000110)
(0000000021)
(0000000102)
(0000000021)
(0010000010)
(0000000022)
(0000000006)
(0000000013)
(0000001010)
(0010000010)
(0010000010)
(0000001100)
(0000000200)
(0200000001)
(0010000002)
p
6=2,3
2
5
6=2,3,5
6=7
3
2
5
6=2,3
3
6=5
6=3
2
5
all
6=5
2
3
6=2,3
3
6=3
3
5
Small degree representations in defining characteristic
6.15. Case A11 , M = 12000
(Recall the remark in 6.5.)
deg
1
12
66
78
142
142
143
220
352
364
495
572
768
780
792
912
924
924
1221
1365
1650
λ
(00000000000)
(00000000001)
(00000000010)
(00000000002)
(10000000001)
(10000000001)
(10000000001)
(00000000100)
(00000000011)
(00000000003)
(00000001000)
(00000000011)
(01000000001)
(01000000001)
(00000010000)
(10000000002)
(00000100000)
(10000000002)
(00000000020)
(00000000004)
(00000000101)
p
all
all
all
all
2
3
6=2,3
all
3
all
all
6=3
11
6=11
all
13
all
6=13
3
all
2
deg
1716
2145
2508
2508
2574
3003
3432
4069
4070
4146
4211
4212
4212
4290
4356
4356
4368
4863
5005
5148
5500
λ
(00000000020)
(00000000101)
(00100000001)
(00100000001)
(00100000001)
(00000000012)
(00000000110)
(01000000010)
(01000000010)
(10000000011)
(01000000010)
(01000000010)
(10000000003)
(10000000003)
(00000000013)
(00000001001)
(00000000005)
(01000000002)
(01000000002)
(00000001001)
(00010000001)
p
6=3
6=2
2
5
6=2,5
all
3
5
2
3
11
6=2,5,11
7
6=7
5
5
all
3
6=3
6=5
3
deg
5720
5797
5939
5940
6642
6654
6720
7656
7656
7656
7788
8074
8514
8580
8580
9009
9504
10296
11220
λ
(00010000001)
(20000000002)
(20000000002)
(20000000002)
(10000000011)
(10000000011)
(10000000011)
(00000000102)
(00000010001)
(00000010001)
(00000000110)
(00000000200)
(00001000001)
(00000000110)
(00000010001)
(00001000001)
(00000100001)
(00000100001)
(00000000021)
p
6=3
7
13
6=7,13
11
13
6=3,11,13
5
2
3
2
3
2
6=2,3
6=2,3
6=2
7
6=7
3
6.16. Case A12 , M = 3000
(Recall the remark in 6.5.)
deg
1
13
78
91
167
168
286
442
λ
(000000000000)
(000000000001)
(000000000010)
(000000000002)
(100000000001)
(100000000001)
(000000000100)
(000000000011)
p
all
all
all
all
13
6=13
all
3
deg
455
715
728
988
988
1001
1157
1170
λ
(000000000003)
(000000001000)
(000000000011)
(010000000001)
(010000000001)
(010000000001)
(100000000002)
(100000000002)
p
all
all
6=3
2
3
6=2,3
7
6=7
deg
1287
1651
1716
1820
2288
2366
λ
(000000010000)
(000000000020)
(000000100000)
(000000000004)
(000000000101)
(000000000020)
λ
(0000000000100)
(0000000000011)
(0000000000003)
(0000000000011)
(0000000001000)
(0100000000001)
(0100000000001)
p
all
3
all
6=3
all
13
6=13
deg
1442
1442
1456
2002
2184
2380
λ
(1000000000002)
(1000000000002)
(1000000000002)
(0000000010000)
(0000000000020)
(0000000000004)
p
all
3
all
all
2
6=3
6.17. Case A13 , M = 3000
(Recall the remark in 6.5.)
deg
1
14
91
105
194
194
195
λ
(0000000000000)
(0000000000001)
(0000000000010)
(0000000000002)
(1000000000001)
(1000000000001)
(1000000000001)
p
all
all
all
all
2
7
6 2,7
=
deg
364
546
560
910
1001
1246
1260
20
p
3
5
6=3,5
all
3
all
Small degree representations in defining characteristic
6.18. Case A14 , M = 3000
(Recall the remark in 6.5.)
deg
1
15
105
120
223
223
λ
(00000000000000)
(00000000000001)
(00000000000010)
(00000000000002)
(10000000000001)
(10000000000001)
p
all
all
all
all
3
5
deg
224
455
665
680
1120
1365
λ
(10000000000001)
(00000000000100)
(00000000000011)
(00000000000003)
(00000000000011)
(00000000001000)
p
6=3,5
all
3
all
6=3
all
deg
1545
1545
1560
1785
2835
λ
(01000000000001)
(01000000000001)
(01000000000001)
(10000000000002)
(00000000000020)
deg
560
800
816
1360
1820
1888
λ
(000000000000100)
(000000000000011)
(000000000000003)
(000000000000011)
(000000000001000)
(010000000000001)
p
all
3
all
6=3
all
3
deg
1888
1904
2144
2160
λ
(010000000000001)
(010000000000001)
(100000000000002)
(100000000000002)
deg
288
680
952
969
1632
λ
(1000000000000001)
(0000000000000100)
(0000000000000011)
(0000000000000003)
(0000000000000011)
p
6=17
all
3
all
6=3
p
2
7
6=2,7
all
3
6.19. Case A15 , M = 3000
(Recall the remark in 6.5.)
deg
1
16
120
136
254
255
λ
(000000000000000)
(000000000000001)
(000000000000010)
(000000000000002)
(100000000000001)
(100000000000001)
p
all
all
all
all
2
6=2
p
5
6=3,5
17
6=17
6.20. Case A16 , M = 3000
(Recall the remark in 6.5.)
deg
1
17
136
153
287
λ
(0000000000000000)
(0000000000000001)
(0000000000000010)
(0000000000000002)
(1000000000000001)
p
all
all
all
all
17
deg
2278
2295
2380
2567
2584
λ
(0100000000000001)
(0100000000000001)
(0000000000001000)
(1000000000000002)
(1000000000000002)
p
2
6=2
all
3
6=3
6.21. Case A17 , M = 3000
(Recall the remark in 6.5.)
deg
1
18
153
171
322
λ
(00000000000000000)
(00000000000000001)
(00000000000000010)
(00000000000000002)
(10000000000000001)
p
all
all
all
all
2
deg
322
323
816
1122
1140
λ
(10000000000000001)
(10000000000000001)
(00000000000000100)
(00000000000000011)
(00000000000000003)
21
p
3
6=2,3
all
3
all
deg
1938
2718
2736
λ
(00000000000000011)
(01000000000000001)
(01000000000000001)
p
6=3
17
6=17
Small degree representations in defining characteristic
6.22. Case B2 , M = 300
deg
1
4
4
5
10
12
13
14
16
20
24
25
25
30
35
35
40
44
52
54
55
56
60
61
64
68
71
76
80
λ
(00)
(01)
(10)
(01)
(20)
(11)
(02)
(02)
(11)
(30)
(12)
(03)
(21)
(03)
(21)
(40)
(12)
(31)
(31)
(04)
(04)
(50)
(14)
(05)
(31)
(22)
(22)
(13)
(13)
p
all
2
all
6=2
all
5
5
6=5
6=5
all
7
7
3
6=7
6=3
all
6=7
7
5
7
6=7
all
11
11
6=5,7
5
7
7
6=7
deg
81
84
84
85
86
91
105
115
116
116
120
126
140
140
140
144
145
149
154
160
164
164
165
174
179
180
180
181
184
λ
(22)
(15)
(60)
(06)
(23)
(05)
(41)
(42)
(33)
(51)
(70)
(06)
(06)
(14)
(32)
(17)
(08)
(42)
(23)
(51)
(34)
(52)
(80)
(07)
(24)
(18)
(33)
(09)
(15)
p
6=5,7
13
all
13
5
6=11
all
5
11
7
all
11
6=11,13
6=11
all
17
17
7
6=5
6=7
13
11
all
13
11
19
7
19
11
deg
199
199
200
204
204
206
220
220
224
231
236
251
256
256
260
260
264
265
271
284
284
285
285
286
294
296
300
λ
(07)
(25)
(71)
(07)
(33)
(24)
(42)
(90)
(15)
(61)
(53)
(25)
(16)
(33)
(24)
(44)
(1,10)
(0,11)
(08)
(08)
(36)
(08)
(43)
(10,0)
(09)
(72)
(52)
p
11
7
11
6=11,13
5
7
6=5,7
all
6=11,13
all
13
13
13
6=5,7,11
6=7,11
7
23
23
13
11
17
6=11,13,17
11
all
17
13
7
6.23. Case B3 , M = 700
deg
1
6
7
8
14
21
26
27
35
40
48
63
64
64
77
104
104
105
112
112
120
132
λ
(000)
(001)
(001)
(100)
(010)
(010)
(002)
(002)
(200)
(101)
(101)
(011)
(011)
(110)
(003)
(110)
(300)
(011)
(110)
(300)
(102)
(020)
p
all
2
6=2
all
2
6=2
7
6=7
all
7
6=7
3
2
5
all
3
5
6=2,3
6=3,5
6=5
3
3
deg
141
155
168
168
168
182
189
189
248
280
293
294
301
304
309
330
344
371
371
371
378
378
λ
(020)
(004)
(020)
(102)
(201)
(004)
(201)
(210)
(120)
(103)
(400)
(400)
(005)
(012)
(012)
(012)
(111)
(005)
(013)
(210)
(005)
(210)
22
p
5
11
6=3,5
6=3
5
6=11
6=5
3
7
11
5
6=5
13
7
3
6=3,7
5
11
5
5
6=11,13
6=3,5
deg
384
448
448
448
472
483
512
518
521
560
560
560
616
616
664
672
672
672
687
693
λ
(111)
(103)
(202)
(310)
(111)
(022)
(111)
(202)
(030)
(104)
(301)
(500)
(021)
(202)
(310)
(021)
(120)
(500)
(006)
(021)
p
3
6=11
5
7
7
5
6=3,5,7
3
7
13
all
7
5
6=3,5
5
3
3
6=7
13
6=3,5
Small degree representations in defining characteristic
6.24. Case B4 , M = 1000
deg
1
8
9
16
26
36
43
44
48
84
112
126
128
147
147
156
λ
(0000)
(0001)
(0001)
(1000)
(0010)
(0010)
(0002)
(0002)
(0100)
(0100)
(1001)
(2000)
(1001)
(0003)
(0011)
(0003)
p
all
2
6=2
all
2
6=2
3
6=3
2
6=2
3
all
6=3
11
3
6=11
deg
160
231
246
304
336
369
406
416
432
448
449
450
451
495
544
558
λ
(0011)
(0011)
(0101)
(1010)
(1100)
(0020)
(0004)
(1010)
(1010)
(1002)
(0004)
(0004)
(0020)
(0020)
(3000)
(0101)
p
all
2
6=2
all
2
6=2
11
6=11
2
2
6=2
13
3
6=13
deg
288
320
320
330
418
429
462
670
749
870
934
935
1088
1143
λ
(10001)
(00011)
(10001)
(01000)
(00011)
(00011)
(20000)
(00101)
(00020)
(00004)
(00004)
(00004)
(10010)
(00020)
p
2
6=2,3
2
7
5
3
13
2
6=2,7
11
11
6=11,13
7
6=3,7
5
7
deg
576
579
594
656
672
752
768
784
798
840
867
874
910
924
957
λ
(1002)
(0110)
(0101)
(1100)
(3000)
(1100)
(1100)
(0110)
(2001)
(2001)
(0012)
(0012)
(0012)
(2001)
(0013)
p
6=11
3
6=2,7
3
6=5
7
6=3,5,7
2
5
3
3
11
6=3,11
6=3,5
5
6.25. Case B5 , M = 2000
deg
1
10
11
32
44
55
64
65
100
164
165
264
264
275
λ
(00000)
(00001)
(00001)
(10000)
(00010)
(00010)
(00002)
(00002)
(00100)
(01000)
(00100)
(00003)
(00011)
(00003)
p
11
2
6=11
6=2
5
6=2,3,5
all
2
3
5
13
6=5,13
3
5
deg
1144
1375
1376
1408
1408
1430
1440
1760
1760
1961
1970
1991
λ
(00020)
(00101)
(10010)
(01001)
(10010)
(00101)
(10002)
(10002)
(11000)
(00012)
(00012)
(00110)
p
6=3,5
3
5
2
6=3,5
6=2,3
13
6=13
5
11
13
3
6.26. Case B6 , M = 4000
deg
1
12
13
64
64
78
89
90
208
286
364
416
429
λ
(000000)
(000001)
(000001)
(000010)
(100000)
(000010)
(000002)
(000002)
(000100)
(000100)
(001000)
(000011)
(000003)
p
all
2
6=2
2
all
6=2
13
6=13
2
6=2
2
3
5
deg
442
560
560
704
715
715
768
1287
1508
1559
1639
1716
1728
λ
(000003)
(000011)
(010000)
(100001)
(000011)
(001000)
(100001)
(010000)
(000101)
(000020)
(000004)
(200000)
(000004)
23
p
6=5
2
2
13
6=2,3
6=2
6=13
6=2
2
3
17
all
5
deg
1729
2185
2275
2847
2925
3392
3808
3838
3849
3849
3927
λ
(000004)
(000020)
(000020)
(000101)
(000101)
(100010)
(001001)
(000012)
(000012)
(000012)
(000012)
p
6=5,17
11
6=3,11
11
6=2,11
11
2
13
3
5
6=3,5,13
Small degree representations in defining characteristic
6.27. Case B7 , M = 5000
deg
1
14
15
90
105
118
118
119
128
336
455
λ
(0000000)
(0000001)
(0000001)
(0000010)
(0000010)
(0000002)
(0000002)
(0000002)
(1000000)
(0000100)
(0000100)
p
all
2
6=2
2
6=2
3
5
6=3,5
all
2
6=2
deg
650
650
665
896
910
1090
1105
1288
1365
1664
1664
λ
(0000003)
(0000011)
(0000003)
(0000011)
(0001000)
(0000011)
(0000011)
(0010000)
(0001000)
(1000001)
(1000001)
p
17
3
6=17
2
2
7
6=2,3,7
2
6=2
3
5
deg
1792
1912
2715
2821
2884
2939
2940
3003
3961
4079
4080
λ
(1000001)
(0100000)
(0000020)
(0000004)
(0000101)
(0000004)
(0000004)
(0010000)
(0000020)
(0000020)
(0000020)
p
6=3,5
2
3
19
2
17
6=17,19
6=2
13
7
6=3,7,13
6.28. Case B8 , M = 7000
deg
1
16
17
118
136
151
152
256
544
680
λ
(00000000)
(00000001)
(00000001)
(00000010)
(00000010)
(00000002)
(00000002)
(10000000)
(00000100)
(00000100)
p
all
2
6=2
2
6=2
17
6=17
all
2
6=2
deg
935
935
952
1344
1582
1615
2380
3808
3840
4096
λ
(00000003)
(00000011)
(00000003)
(00000011)
(00001000)
(00000011)
(00001000)
(00010000)
(10000001)
(10000001)
p
19
3
6=19
2
2
6=2,3
6=2
2
17
6=17
deg
4251
4488
4540
4691
4692
5066
6188
6528
6631
6783
λ
(00000020)
(00100000)
(00000004)
(00000004)
(00000004)
(00000101)
(00010000)
(01000000)
(00000020)
(00000020)
deg
780
969
1273
1292
1311
1920
2261
2906
λ
(000000100)
(000000100)
(000000011)
(000000003)
(000000003)
(000000011)
(000000011)
(000001000)
p
2
6=2
3
7
6=7
2
6=2,3
2
deg
3876
6763
6936
6972
7124
7125
λ
(000001000)
(000000020)
(000000004)
(000010000)
(000000004)
(000000004)
λ
(0000000002)
(1000000000)
(0000000100)
(0000000100)
(0000000003)
(0000000011)
(0000000003)
p
6=3,7
all
2
6=2
23
3
6=23
deg
2640
3038
3059
4466
5985
9954
λ
(0000000011)
(0000000011)
(0000000011)
(0000001000)
(0000001000)
(0000000020)
λ
(00000000002)
(00000000100)
(00000000100)
(10000000000)
(00000000003)
(00000000011)
p
6=23
2
6=2
all
5
3
deg
2277
3520
4002
4025
7084
8855
λ
(00000000003)
(00000000011)
(00000000011)
(00000000011)
(00000001000)
(00000001000)
p
3
2
7
19
6=7,19
2
6=2
2
5
6=3,5
6.29. Case B9 , M = 8000
deg
1
18
19
152
171
188
189
512
λ
(000000000)
(000000001)
(000000001)
(000000010)
(000000010)
(000000002)
(000000002)
(100000000)
p
all
2
6=2
2
6=2
19
6=19
all
p
6=2
3
23
2
7
6 7,23
=
6.30. Case B10 , M = 10000
deg
1
20
21
188
210
229
229
λ
(0000000000)
(0000000001)
(0000000001)
(0000000010)
(0000000010)
(0000000002)
(0000000002)
p
all
2
6=2
2
6=2
3
7
deg
230
1024
1120
1330
1729
1729
1750
p
2
5
6=2,3,5
2
6=2
3
6.31. Case B11 , M = 12000
deg
1
22
23
230
253
274
λ
(00000000000)
(00000000001)
(00000000001)
(00000000010)
(00000000010)
(00000000002)
p
all
2
6=2
2
6=2
23
deg
275
1496
1771
2048
2254
2254
24
p
6=5
2
11
6=2,3,11
2
6=2
Small degree representations in defining characteristic
6.32. Case C3 , M = 1000
deg
1
6
8
13
14
14
21
48
50
56
57
58
62
63
64
70
84
89
90
112
126
126
158
171
λ
(000)
(001)
(100)
(010)
(010)
(100)
(002)
(101)
(011)
(003)
(101)
(011)
(110)
(200)
(011)
(101)
(200)
(020)
(020)
(110)
(004)
(110)
(102)
(210)
p
all
all
2
3
6=3
6=2
all
2
3
all
3
7
5
5
6=3,7
6=2,3
6=5
7
6=7
2
all
6=2,5
7
7
deg
172
189
202
216
246
252
286
295
309
316
316
330
350
358
378
385
423
426
448
462
512
525
552
573
λ
(300)
(012)
(102)
(102)
(013)
(005)
(021)
(030)
(111)
(120)
(201)
(300)
(021)
(111)
(201)
(030)
(111)
(022)
(013)
(006)
(111)
(103)
(120)
(210)
p
7
all
5
6=5,7
5
all
3
5
5
3
5
6=7
6=3
3
6=5
6=5
7
5
6=5
all
6=3,5,7
all
5
5
deg
594
616
623
665
666
666
786
792
798
813
840
854
861
875
903
924
924
924
938
942
951
951
994
λ
(210)
(120)
(202)
(410)
(031)
(500)
(015)
(007)
(014)
(301)
(130)
(211)
(301)
(400)
(022)
(014)
(022)
(112)
(400)
(104)
(040)
(202)
(031)
p
6=5,7
6=3,5
3
11
5
11
7
all
5
5
7
7
7
7
3
6=5
6=3,5
5
5
7
5
5
11
6.33. Case C4 , M = 2000
deg
1
8
16
26
27
36
40
41
42
48
112
120
128
160
240
246
266
279
281
288
308
λ
(0000)
(0001)
(1000)
(0010)
(0010)
(0002)
(0100)
(1000)
(1000)
(0100)
(0011)
(0003)
(1001)
(0011)
(1001)
(0101)
(0020)
(0101)
(0020)
(1001)
(0020)
p
all
all
2
2
6=2
all
3
3
6=2,3
6=3
3
all
2
6=3
7
2
3
3
5
6=2,7
6=3,5
deg
312
313
315
330
416
504
513
558
593
594
594
632
744
765
768
784
784
784
789
792
792
λ
(1100)
(2000)
(0101)
(0004)
(1010)
(0110)
(1010)
(0012)
(2000)
(0012)
(2000)
(0110)
(0110)
(1010)
(1100)
(0013)
(0110)
(0200)
(0200)
(0005)
(0110)
25
p
5
5
6=2,3
all
2
3
3
5
7
6=5
6=5,7
7
5
7
2
5
2
3
7
all
6 2,3,5,7
=
deg
792
825
944
1016
1048
1056
1072
1114
1155
1200
1201
1232
1352
1504
1512
1608
1652
1716
1728
1891
λ
(1010)
(0200)
(0102)
(1100)
(1100)
(1100)
(0102)
(1002)
(1002)
(2100)
(3000)
(0102)
(0021)
(0021)
(0021)
(0013)
(0022)
(0006)
(0013)
(0111)
p
6=2,3,7
6=3,7
5
3
7
6=2,3,5,7
3
3
6=3
7
7
6=3,5
11
5
6=5,11
11
5
all
6=5,11
3
Small degree representations in defining characteristic
6.34. Case C5 , M = 2500
deg
1
10
32
43
44
55
100
110
121
122
132
164
165
λ
(00000)
(00001)
(10000)
(00010)
(00010)
(00002)
(00100)
(00100)
(01000)
(10000)
(10000)
(01000)
(01000)
p
all
all
2
5
6=5
all
2
6=2
3
3
6=2,3
2
6=2,3
deg
210
220
310
320
320
615
670
715
779
780
848
891
1088
6.35. Case C6 , M = 4000
deg
1
12
64
64
64
65
78
196
208
352
364
364
364
λ
(000000)
(000001)
(000010)
(000010)
(100000)
(000010)
(000002)
(000100)
(000100)
(000011)
(000003)
(001000)
(010000)
p
all
all
2
3
2
6=2,3
all
5
6=5
3
all
2
3
deg
365
428
429
429
548
560
560
572
768
1221
1365
1508
1585
λ
(100000)
(001000)
(001000)
(100000)
(000011)
(000011)
(010000)
(010000)
(100001)
(000020)
(000004)
(000101)
(000020)
deg
882
896
909
910
1093
1094
1288
1430
1624
1638
1792
1912
λ
(0000011)
(0000011)
(0001000)
(0001000)
(0100000)
(1000000)
(0010000)
(1000000)
(0010000)
(0010000)
(1000001)
(0100000)
deg
1582
1699
1700
3264
3280
3281
3620
3792
3808
3876
4096
4488
4861
λ
(00001000)
(00001000)
(00001000)
(00010000)
(01000000)
(10000000)
(00000020)
(00010000)
(00010000)
(00000004)
(10000001)
(00100000)
(10000000)
6.36. Case C7 , M = 6000
deg
1
14
89
90
105
128
336
336
350
546
560
820
λ
(0000000)
(0000001)
(0000010)
(0000010)
(0000002)
(1000000)
(0000100)
(0000100)
(0000100)
(0000011)
(0000003)
(0001000)
p
all
all
7
6=7
all
2
2
3
6=2,3
3
all
5
6.37. Case C8 , M = 10000
deg
1
16
118
119
136
256
528
544
800
816
1328
1344
1581
λ
(00000000)
(00000001)
(00000010)
(00000010)
(00000002)
(10000000)
(00000100)
(00000100)
(00000011)
(00000003)
(00000011)
(00000011)
(00001000)
p
all
all
2
6=2
all
2
7
6=7
3
all
17
6=3,17
3
λ
(00011)
(00003)
(00011)
(00011)
(10001)
(00020)
(00101)
(00004)
(00020)
(00020)
(00101)
(00101)
(01001)
26
p
3
all
11
6=3,11
2
3
2
all
11
6=3,11
5
6=2,5
3
deg
1099
1155
1276
1375
1408
1408
1430
1452
1562
1563
1992
2002
λ
(10001)
(10001)
(01001)
(00012)
(01001)
(10010)
(00012)
(00110)
(11000)
(20000)
(00013)
(00005)
p
3
5
6=2,5
6=2,3
13
6=3,13
2
6=2,3
2
3
all
2
7
deg
1649
1650
1924
1938
2002
2847
2925
3432
3638
3652
3796
3808
λ
(000020)
(000020)
(000101)
(000101)
(000101)
(000012)
(000012)
(000110)
(010001)
(100001)
(001001)
(001001)
p
5
6=3,5
3
6=3,5
3
3
2
6=2,3
5
6=2,5
2
2
deg
2001
2002
2184
2380
2884
3093
3094
3795
3811
3900
5355
λ
(0100000)
(0100000)
(0000020)
(0000004)
(0000101)
(0000020)
(0000020)
(0000101)
(0000101)
(0000101)
(0000012)
p
2
7
6=2,3,7
5
3
3
3
3
6=3,5
all
2
2
5
deg
4862
5066
5319
5320
6069
6188
6528
6749
6885
7056
7072
8908
9044
p
3
6=2,3
5
3
6=3,5
2
6=3
3
5
5
5
all
p
13
6=3,7,13
5
3
6=2,3,5
7
6=7
3
3
3
5
2
λ
(10000000)
(00000101)
(00000020)
(00000020)
(00100000)
(00100000)
(01000000)
(00000101)
(00000101)
(01000000)
(01000000)
(00000012)
(00000012)
p
5
6=2,3,5
3
all
2
5
6=3,5
3
7
6=2,3,7
all
p
6=2,3,5
2
17
6=3,17
5
6=2,5
2
7
6=2,7
5
6=2,3,5
3
6=3
Small degree representations in defining characteristic
6.38. Case C9 , M = 10000
deg
1
18
151
152
171
512
780
798
1122
1140
λ
(000000000)
(000000001)
(000000010)
(000000010)
(000000002)
(100000000)
(000000100)
(000000100)
(000000011)
(000000003)
p
all
all
3
6=3
all
2
2
6=2
3
all
deg
1902
1920
2755
2906
2907
5661
5985
6954
6972
7734
6.39. Case C10 , M = 10000
deg
1
20
188
188
189
210
λ
(0000000000)
(0000000001)
(0000000010)
(0000000010)
(0000000010)
(0000000002)
p
all
all
2
5
6=2,5
all
deg
1024
1100
1120
1520
1540
2620
6.40. Case C11 , M = 12000
deg
1
22
229
230
253
1496
λ
(00000000000)
(00000000001)
(00000000010)
(00000000010)
(00000000002)
(00000000100)
p
all
all
11
6=11
all
2
deg
1496
1518
2002
2024
2048
3498
λ
(000000011)
(000000011)
(000001000)
(000001000)
(000001000)
(000000020)
(000000004)
(000010000)
(000010000)
(000010000)
λ
(1000000000)
(0000000100)
(0000000100)
(0000000011)
(0000000003)
(0000000011)
λ
(00000000100)
(00000000100)
(00000000011)
(00000000003)
(10000000000)
(00000000011)
27
p
19
6=3,19
7
2
6=2,7
3
all
3
2
7
p
2
3
6=3
3
all
7
p
5
6=2,5
3
all
2
23
deg
7752
8226
8416
8567
8568
9216
9841
9842
deg
2640
4466
4654
4655
8455
8855
deg
3520
6854
7083
7084
λ
(000010000)
(000000101)
(000000020)
(000000020)
(000000020)
(100000001)
(010000000)
(100000000)
p
6=2,3,7
2
5
19
6=3,5,19
2
3
3
λ
(0000000011)
(0000001000)
(0000001000)
(0000001000)
(0000000020)
(0000000004)
λ
(00000000011)
(00000001000)
(00000001000)
(00000001000)
p
6=3,7
2
3
6=2,3
3
all
p
6=3,23
3
5
6=3,5
Small degree representations in defining characteristic
6.41. Case D4 , M = 2000
Here, because of the symmetry of the Dynkin diagram, a weight (a1 , a2 , a3 , a4 ) and all
those which are got from this by permuting a1 , a2 and a4 lead to representations of same
degree. But for better readability we include all of these weights in this table.
deg
1
8
8
8
26
28
35
35
35
48
48
48
56
56
56
104
104
104
104
104
104
112
112
112
152
152
152
160
160
160
168
168
168
168
168
168
195
224
224
224
224
224
224
246
293
293
293
294
294
294
299
300
λ
(0000)
(0001)
(0100)
(1000)
(0010)
(0010)
(0002)
(0200)
(2000)
(0101)
(1001)
(1100)
(0101)
(1001)
(1100)
(0003)
(0011)
(0110)
(0300)
(1010)
(3000)
(0003)
(0300)
(3000)
(0011)
(0110)
(1010)
(0011)
(0110)
(1010)
(0102)
(0201)
(1002)
(1200)
(2001)
(2100)
(0020)
(0102)
(0201)
(1002)
(1200)
(2001)
(2100)
(1101)
(0004)
(0400)
(4000)
(0004)
(0400)
(4000)
(0020)
(0020)
p
all
all
all
all
2
6=2
all
all
all
2
2
2
6=2
6=2
6=2
5
3
3
5
3
5
6=5
6=5
6=5
7
7
7
6=3,7
6=3,7
6=3,7
5
5
5
5
5
5
3
6=5
6=5
6=5
6=5
6=5
6=5
2
5
5
5
6=5
6=5
6=5
7
6=3,7
deg
322
350
384
384
384
518
518
518
539
539
539
560
560
560
567
567
567
664
664
664
664
664
664
664
664
664
672
672
672
672
672
672
672
672
672
680
680
680
784
784
784
805
805
805
840
840
840
840
840
840
904
904
λ
(1101)
(1101)
(0111)
(1011)
(1110)
(0202)
(2002)
(2200)
(0012)
(0210)
(2010)
(0005)
(0500)
(5000)
(0012)
(0210)
(2010)
(0013)
(0103)
(0301)
(0310)
(1003)
(1300)
(3001)
(3010)
(3100)
(0005)
(0103)
(0301)
(0500)
(1003)
(1300)
(3001)
(3100)
(5000)
(0111)
(1011)
(1110)
(0111)
(1011)
(1110)
(0202)
(2002)
(2200)
(0111)
(0202)
(1011)
(1110)
(2002)
(2200)
(1102)
(1201)
28
p
3
6=2,3
3
3
3
3
3
3
5
5
5
7
7
7
6=5
6=5
6=5
5
5
5
5
5
5
5
5
5
6=7
6=5
6=5
6=7
6=5
6=5
6=5
6=5
6=7
5
5
5
2
2
2
5
5
5
6=2,3,5
6=3,5
6=2,3,5
6=2,3,5
6=3,5
6=3,5
5
5
deg
904
1008
1008
1008
1008
1008
1008
1144
1144
1144
1176
1176
1176
1256
1256
1256
1256
1256
1256
1296
1296
1296
1322
1322
1322
1351
1351
1351
1386
1386
1386
1400
1400
1400
1568
1568
1568
1680
1680
1680
1680
1680
1680
1841
1896
1896
1896
1896
1896
1896
1925
λ
(2101)
(0104)
(0401)
(1004)
(1400)
(4001)
(4100)
(1102)
(1201)
(2101)
(0021)
(0120)
(1020)
(0203)
(0302)
(2003)
(2300)
(3002)
(3200)
(1102)
(1201)
(2101)
(0022)
(0220)
(2020)
(0006)
(0600)
(6000)
(0006)
(0600)
(6000)
(0021)
(0120)
(1020)
(0013)
(0310)
(3010)
(0104)
(0401)
(1004)
(1400)
(4001)
(4100)
(1111)
(0112)
(0211)
(1012)
(1210)
(2011)
(2110)
(0030)
p
5
7
7
7
7
7
7
7
7
7
3
3
3
7
7
7
7
7
7
6=5,7
6=5,7
6=5,7
5
5
5
7
7
7
6=7
6=7
6=7
6=3
6=3
6=3
6=5
6=5
6=5
6=7
6=7
6=7
6=7
6=7
6=7
3
3
3
3
3
3
3
all
Small degree representations in defining characteristic
6.42. Case D5 , M = 3000
deg
1
10
16
16
44
45
53
54
100
120
126
126
128
128
144
144
164
190
210
210
320
416
416
544
544
544
544
559
560
560
λ
(00000)
(00001)
(01000)
(10000)
(00010)
(00010)
(00002)
(00002)
(00100)
(00100)
(02000)
(20000)
(01001)
(10001)
(01001)
(10001)
(11000)
(00011)
(00003)
(11000)
(00011)
(01010)
(10010)
(01010)
(03000)
(10010)
(30000)
(00020)
(01010)
(10010)
p
all
all
all
all
2
6=2
5
6=5
2
6=2
all
all
5
5
6=5
6=5
2
3
all
6=2
6=3
2
2
3
5
3
5
3
6=2,3
6=2,3
deg
576
576
606
656
656
660
670
672
672
720
720
770
840
840
880
880
945
1050
1050
1056
1056
1184
1184
1200
1200
1242
1333
1341
1386
1408
λ
(01002)
(10002)
(00004)
(01100)
(10100)
(00004)
(00101)
(03000)
(30000)
(01002)
(10002)
(00020)
(02001)
(20001)
(12000)
(21000)
(00101)
(02001)
(20001)
(01100)
(10100)
(01100)
(10100)
(01100)
(10100)
(00110)
(00012)
(00012)
(00012)
(11001)
deg
462
462
495
548
560
560
792
1143
1286
1287
1376
1376
1508
1561
1637
1638
1696
1696
1728
1728
λ
(020000)
(200000)
(001000)
(000011)
(000011)
(110000)
(110000)
(000020)
(000004)
(000004)
(010010)
(100010)
(000101)
(000020)
(000020)
(000020)
(010010)
(100010)
(010010)
(100010)
p
3
3
7
3
3
6=7
2
6=5
6=5
6=3
6=3
6=3
3
3
5
5
6=2
6=3
6=3
7
7
2
2
6=2,3,7
6=2,3,7
3
5
3
6=3,5
2
deg
1424
1424
1440
1440
1476
1608
1676
1728
1772
1772
1782
1920
1920
2094
2094
2464
2464
2640
2640
2650
2708
2719
2719
2772
2772
2836
2970
2976
2976
λ
(12000)
(21000)
(12000)
(21000)
(11001)
(11001)
(00200)
(11001)
(00005)
(00013)
(00005)
(01003)
(10003)
(02010)
(20010)
(01011)
(10011)
(01003)
(10003)
(00110)
(00110)
(04000)
(40000)
(04000)
(40000)
(00102)
(00110)
(01011)
(10011)
p
7
7
6=5,7
6=5,7
5
7
3
6=2,5,7
7
5
6=7
7
7
5
5
3
3
6=7
6=7
7
2
5
5
6=5
6=5
5
6=2,3,7
2
2
6.43. Case D6 , M = 4000
deg
1
12
32
32
64
66
76
77
208
220
320
320
320
320
340
340
352
352
352
364
λ
(000000)
(000001)
(010000)
(100000)
(000010)
(000010)
(000002)
(000002)
(000100)
(000100)
(010001)
(010001)
(100001)
(100001)
(000003)
(000011)
(000003)
(010001)
(100001)
(001000)
p
all
all
all
all
2
6=2
3
6=3
2
6=2
2
3
2
3
7
3
6=7
6=2,3
6=2,3
2
29
p
all
all
6=2
11
6=3,11
2
6=2
3
7
6=7
5
5
2
5
11
6=3,5,11
11
11
6=5,11
6=5,11
deg
1760
1760
2013
2079
2112
2112
2740
2784
2794
2848
2848
2860
3200
3200
3808
3960
3960
3992
λ
(010002)
(100002)
(000101)
(000101)
(010002)
(100002)
(000110)
(000012)
(000012)
(030000)
(300000)
(000012)
(010100)
(100100)
(001001)
(020001)
(200001)
(000013)
p
7
7
5
6=2,5
6=7
6=7
3
3
7
5
5
6=3,7
2
2
2
7
7
5
Small degree representations in defining characteristic
6.44. Case D7 , M = 5000
deg
1
14
64
64
90
91
103
104
336
364
532
546
768
768
λ
(0000000)
(0000001)
(0100000)
(1000000)
(0000010)
(0000010)
(0000002)
(0000002)
(0000100)
(0000100)
(0000011)
(0000003)
(0100001)
(1000001)
p
all
all
all
all
2
6=2
7
6=7
2
6=2
3
all
7
7
deg
832
832
882
896
910
1001
1288
1716
1716
1912
1975
2002
2275
2884
λ
(0100001)
(1000001)
(0000011)
(0000011)
(0001000)
(0001000)
(0010000)
(0200000)
(2000000)
(1100000)
(0000020)
(0010000)
(0000004)
(0000101)
p
6=7
6=7
13
6=3,13
2
6=2
2
all
all
2
3
6=2
all
2
deg
3003
3079
3080
3913
4004
4096
4096
4096
4096
4864
4864
4928
4928
λ
(1100000)
(0000020)
(0000020)
(0000101)
(0000101)
(0100010)
(0100010)
(1000010)
(1000010)
(0100010)
(1000010)
(0100010)
(1000010)
p
6=2
13
6=3,13
3
6=2,3
2
3
2
3
13
13
6=2,3,13
6=2,3,13
6.45. Case D8 , M = 10000
deg
1
16
118
120
128
128
135
544
560
768
800
1328
1344
λ
(00000000)
(00000001)
(00000010)
(00000010)
(01000000)
(10000000)
(00000002)
(00000100)
(00000100)
(00000011)
(00000003)
(00000011)
(00000011)
p
all
all
2
6=2
all
all
all
2
6=2
3
all
5
6=3,5
deg
1582
1792
1792
1820
1920
1920
3483
3605
3740
3808
4368
4488
5066
λ
(00001000)
(01000001)
(10000001)
(00001000)
(01000001)
(10000001)
(00000020)
(00000004)
(00000004)
(00010000)
(00010000)
(00100000)
(00000101)
p
2
2
2
6=2
6=2
6=2
3
5
6=5
2
6=2
2
2
deg
5169
5303
5304
6435
6435
6528
6900
7020
8008
8805
8925
λ
(00000020)
(00000020)
(00000020)
(02000000)
(20000000)
(11000000)
(00000101)
(00000101)
(00100000)
(00000012)
(00000012)
p
7
5
6=3,5,7
all
all
2
7
6=2,7
6=2
3
6=3
6.46. Case D9 , M = 15000
deg
1
18
152
153
169
170
256
256
780
816
1104
1104
λ
(000000000)
(000000001)
(000000010)
(000000010)
(000000002)
(000000002)
(010000000)
(100000000)
(000000100)
(000000100)
(000000003)
(000000011)
p
all
all
2
6=2
3
6=3
all
all
2
6=2
5
3
deg
1122
1902
1920
2906
3060
4096
4096
4352
4352
5490
5644
5813
λ
(000000003)
(000000011)
(000000011)
(000001000)
(000001000)
(010000001)
(100000001)
(010000001)
(100000001)
(000000020)
(000000004)
(000000004)
p
6=5
17
6=3,17
2
6=2
3
3
6=3
6=3
3
11
5
deg
5814
6972
8226
8549
8550
8568
11475
14043
14059
14212
λ
(000000004)
(000010000)
(000000101)
(000000020)
(000000020)
(000010000)
(000000101)
(000000012)
(000000012)
(000000012)
deg
1500
1520
2620
2640
4466
4845
8036
8644
8645
9216
9216
λ
(0000000011)
(0000000003)
(0000000011)
(0000000011)
(0000001000)
(0000001000)
(0000000020)
(0000000004)
(0000000004)
(0100000001)
(0100000001)
p
3
6=11
19
6=3,19
2
6=2
3
11
6=11
2
5
deg
9216
9216
9728
9728
12712
13089
13090
14344
15504
17575
17765
λ
(1000000001)
(1000000001)
(0100000001)
(1000000001)
(0000000101)
(0000000020)
(0000000020)
(0000010000)
(0000010000)
(0000000101)
(0000000101)
p
6=5,11
2
2
17
6=3,17
6=2
6=2
3
5
6=3,5
6.47. Case D10 , M = 18000
deg
1
20
188
190
208
209
512
512
1120
1140
1500
λ
(0000000000)
(0000000001)
(0000000010)
(0000000010)
(0000000002)
(0000000002)
(0100000000)
(1000000000)
(0000000100)
(0000000100)
(0000000003)
p
all
all
2
6=2
5
6=5
all
all
2
6=2
11
30
p
2
5
6=2,5
6=2,5
2
19
6=3,19
2
6=2
3
6=2,3
Small degree representations in defining characteristic
6.48. Case D11 , M = 20000
deg
1
22
230
231
251
252
1024
1024
λ
(00000000000)
(00000000001)
(00000000010)
(00000000010)
(00000000002)
(00000000002)
(01000000000)
(10000000000)
p
all
all
2
6=2
11
6=11
all
all
λ
(00000000100)
(00000000100)
(00000000011)
(00000000003)
(00000000011)
(00000000011)
(00000001000)
(00000001000)
deg
1496
1540
1958
2002
3498
3520
7084
7315
p
2
6=2
3
all
7
6=3,7
2
6=2
deg
11912
12145
12397
18744
18976
19227
19228
λ
(00000000020)
(00000000004)
(00000000004)
(00000000101)
(00000000020)
(00000000020)
(00000000020)
6.49. Case G2 , M = 500
deg
1
6
7
7
14
26
27
27
38
49
64
77
77
λ
(00)
(01)
(01)
(10)
(10)
(02)
(02)
(20)
(11)
(11)
(11)
(03)
(20)
p
all
2
6=2
3
6=3
7
6=7
3
7
3
6=3,7
all
6=3
deg
97
125
148
155
182
189
189
196
244
248
259
267
273
λ
(21)
(12)
(30)
(04)
(04)
(12)
(21)
(30)
(31)
(21)
(13)
(40)
(30)
p
5
11
11
11
6=11
6=11
3
5
13
7
13
7
6=5,11
deg
286
295
301
371
371
378
434
448
469
481
483
489
λ
(21)
(22)
(05)
(05)
(13)
(05)
(13)
(13)
(14)
(22)
(22)
(40)
p
6=3,5,7
11
13
11
5
6=11,13
11
6=5,11,13
5
7
5
13
6.50. Case F4 , M = 12000
deg
1
25
26
26
52
196
246
246
273
298
323
324
676
λ
(0000)
(0001)
(0001)
(1000)
(1000)
(0010)
(0010)
(0100)
(0010)
(0002)
(0002)
(0002)
(1001)
p
all
3
6=3
2
6=2
3
2
2
6=2,3
7
13
6=7,13
2
deg
755
1053
1053
1222
1274
2404
2651
2652
2991
3773
4096
4096
4380
λ
(2000)
(1001)
(2000)
(0100)
(0100)
(0011)
(0003)
(0003)
(0011)
(0011)
(0011)
(1100)
(1010)
31
p
7
6=2
6=7
3
6=2,3
3
7
6=7
7
13
6=3,7,13
2
7
deg
5369
6396
6396
6707
7371
8424
9477
10829
11102
11739
11907
λ
(1100)
(0101)
(1010)
(1002)
(1010)
(1010)
(2001)
(1002)
(3000)
(0110)
(0101)
p
3
2
2
3
3
6=2,3,7
5
6=3
5
5
3
p
3
13
6=13
2
5
7
6 3,5,7
=
Small degree representations in defining characteristic
6.51. Case E6 , M = 50000
deg
1
27
27
77
78
324
324
324
324
351
351
351
351
572
572
650
1377
1377
1701
1701
1702
1728
1728
2404
2404
2429
2430
2771
2925
3002
3002
3003
λ
(000000)
(000001)
(100000)
(010000)
(010000)
(000002)
(000010)
(001000)
(200000)
(000002)
(000010)
(001000)
(200000)
(100001)
(100001)
(100001)
(010001)
(110000)
(010001)
(110000)
(000100)
(010001)
(110000)
(000011)
(101000)
(020000)
(020000)
(000100)
(000100)
(000003)
(300000)
(000003)
p
all
all
all
3
6=3
5
2
2
5
6=5
6=2
6=2
6=5
2
3
6=2,3
5
5
13
13
2
6=5,13
6=5,13
3
3
13
6=13
3
6=2,3
5
5
6=5
deg
3003
5643
5643
5746
5746
5824
5824
5994
5994
6966
6966
7371
7371
7371
7371
7722
7722
9828
9828
11934
11934
13311
13311
15822
15822
17550
17550
18225
18225
18954
18954
19305
6.52. Case E7 , M = 100000
deg
1
56
132
133
856
912
1274
1330
1463
1538
1539
5568
5832
λ
(0000000)
(0000001)
(1000000)
(1000000)
(0100000)
(0100000)
(0000010)
(0000002)
(0000002)
(0000010)
(0000010)
(1000001)
(2000000)
p
all
all
2
6=2
3
6=3
2
3
6=3
7
6=2,7
7
5
deg
6424
6480
7106
7370
7371
8512
8645
18752
21184
24264
24264
24320
25896
6.53. Case E8 , M = 100000
deg
1
248
3626
3875
λ
(00000000)
(00000001)
(10000000)
(10000000)
p
all
all
2
6=2
deg
23125
26504
26999
27000
λ
(300000)
(001001)
(100010)
(000011)
(101000)
(000011)
(101000)
(100002)
(200001)
(001001)
(100010)
(001001)
(100002)
(100010)
(200001)
(100002)
(200001)
(010010)
(011000)
(010002)
(210000)
(000004)
(400000)
(010010)
(011000)
(010010)
(011000)
(000020)
(002000)
(010002)
(210000)
(000004)
p
6=5
5
5
5
5
6=3,5
6=3,5
7
7
2
2
6=2,5
5
6=2,5
5
6=5,7
6=5,7
2
2
3
3
7
7
3
3
6=2,3
6=2,3
3
3
7
7
6=7
deg
19305
19305
19305
20579
23179
26244
26244
28782
28782
31668
32319
33021
33021
34099
34398
34398
34749
35242
40831
43758
44955
44955
46332
46332
46656
46656
46683
46683
46683
46683
λ
(010002)
(210000)
(400000)
(030000)
(110001)
(000101)
(100100)
(020001)
(120000)
(110001)
(110001)
(000020)
(002000)
(110001)
(000020)
(002000)
(110001)
(001010)
(010100)
(030000)
(020001)
(120000)
(020001)
(120000)
(001002)
(200010)
(000012)
(000012)
(201000)
(201000)
λ
(1000001)
(1000001)
(0010000)
(2000000)
(2000000)
(0010000)
(0010000)
(0000003)
(0000100)
(0000003)
(0000011)
(0000003)
(0000100)
p
19
6=7,19
2
19
6=5,19
3
6=2,3
7
2
11
3
6=7,11
3
deg
27664
30704
30779
39217
40755
44592
50160
51072
57608
79704
86128
86184
λ
(0000100)
(0100001)
(0100001)
(0100001)
(0100001)
(0000011)
(0000011)
(0000011)
(1100000)
(1100000)
(1100000)
(1100000)
λ
(00000002)
(00000010)
(00000002)
(00000002)
p
7
2
31
6 7,31
=
deg
30132
30132
30380
λ
(00000010)
(00000010)
(00000010)
p
6=3,7
6=3,7
6=7
5
5
2
2
3
3
2
3
5
5
7
6=3,5
6=3,5
6=2,3,5,7
2
3
6=5
5
5
6=3,5
6=3,5
3
3
3
5
3
5
p
6=2,3
2
3
7
6=2,3,7
11
2
6=2,3,11
5
13
2
6=2,5,13
p
3
5
6 2,3,5
=
References
1. D. J. B ENSON, Representations and Cohomology I, vol. 30 of Cambridge studies in
advanced mathematics (Cambridge University Press, Cambridge, 1991). 9
32
Small degree representations in defining characteristic
2. N. B URGOYNE, ‘Modular representations of some finite groups.’ ‘Representation
Theory of Finite Groups and Related Topics,’ (AMS, Providence, 1971), vol. XXI of
Proc. Symp. Pure Math. pp. 13–17. 3
3. R.W. C ARTER, Simple Groups of Lie Type (A Wiley-Interscience publication, London, 1972). 4
4. C. W. C URTIS and I. R EINER, Methods of representation theory, vol. I, II (Wiley,
New York, 1981/1987). 9
5. M. G ECK, G. H ISS, F. L ÜBECK, G. M ALLE and G. P FEIFFER, ‘CHEVIE – A system
for computing and processing generic character tables for finite groups of Lie type,
Weyl groups and Hecke algebras.’ AAECC 7 (1996) 175–210. 4, 10
6. P. B. G ILKEY and G. M. S EITZ, ‘Some representations of exceptional Lie algebras.’
Geometriae Dedicata 25 (1988) 407–416. 1, 3, 4
7. G. H ISS and G. M ALLE, ‘Low-dimensional representations of quasi-simple groups.’
J. Comput. Math., London Mathematical Society 4 (2001) 22–63. 1, 10
8. J. E. H UMPHREYS, Introduction to Lie Algebras and Representation Theory.
(Springer Verlag, New York-Heidelberg-Berlin, 1972). 3, 4, 5, 7, 9
9. J. E. H UMPHREYS, ‘Modular representations in finite groups of Lie type.’ ‘Finite
Simple Groups II,’ (ed. M. J. C OLLINS) (Academic Press, London, 1980) pp. 259–
290. 2
10. C. JANSEN , K. L UX , R. PARKER AND R. W ILSON An Atlas of Brauer Characters.
(The Clarendon Press Oxford University Press, New York, 1995). 7
11. J. C. JANTZEN, ‘Darstellungen halbeinfacher Gruppen und kontravariante Formen.’
J. Reine Angew. Math. 290 (1977) 117–141. 1
12. J. C. JANTZEN, Representations of Algebraic Groups. No. 131 in Pure and Applied
Mathematics (Academic Press, London, 1987). 2, 4
13. P. K LEIDMAN and M. L IEBECK, The subgroup structure of the finite classical
groups. No. 129 in LMS Lecture Note Series (Cambridge University Press, 1990).
1, 9
14. F. L ÜBECK, ‘On the computation of elementary divisors of integer matrices.’ Journal
of Symbolic Computation (to appear). 4
15. R. V. M OODY and J. PATERA ‘Fast recursion formula for weight multiplicities’ Bull.
Amer. Math. Soc. (N.S.) 7 (1982) 237–242. 5
16. A. A. P REMET, ‘Weights of infinitesimally irreducible representations of Chevalley
groups over a field of prime characteristic.’ Math. USSR Sb. 61 (1988) 167–183.
(translated from Russian). 5
17. M ARTIN S CHÖNERT and OTHERS, GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule,
Aachen, Germany, 3rd edn., (1993–1997). 4
18. R. S TEINBERG, ‘Lectures on Chevalley groups.’ Yale University, (1967). 11
19. R. S TEINBERG, ‘Endomorphisms of linear algebraic groups.’ No. 80 of AMS memoirs (1968). 3
33
Small degree representations in defining characteristic
Frank Lübeck
RWTH Aachen
Lehrstuhl D für Mathematik
Templergraben 64
52062 Aachen
Germany
E-mail: Frank.Luebeck@Math.RWTH-Aachen.De
WWW: http://www.math.rwth-aachen.de/˜Frank.Luebeck/
34