For each irreducible module Xi Nanhua defined an element which generated this module. We use this element to construct a certain basis for and then compute dim , determine its formal characters in this paper. In order to obtain faster speed we modify the algorithm to compute the irreducible characters.
The determination of all irreducible characters is a big theme in the modular representations of algebraic groups and related finite groups of Lie type. But so far only a little is known concerning it in the case when the characteristic of the base field is less than the Coxeter number.
Gilkey-Seitz gave an algorithm to compute part of characters of
An element
39.1, p. 304] and [[
Let
be the set of simple roots of
Let
Let
For
Let
Then the irreducible
For
Then the Steinberg tensor product theorem tells us that
Therefore we can determine all the characters
Recall the strong linkage principle in [
Let
Then
gebra over
tively. They are generated by
bra, and
algebra of
particular,
with
Let
As a special case of [[
Theorem 1 Assume that
(i) The element
(ii) Let
(iii) As a
By abuse of notations, the images in
From now on we shall assume that
Now we can obtain our main theorems. Let
for all
form bases of
with
from
Let us mention our computation of
with
For example, we assume that
It is easy to see that
For
and then we compute the rank of the set
When
Then the Steinberg tensor product theorem tells us that
Therefore, we can determine all characters
Therefore, from the two matrices
We list the matrix
In paper [
For example, suppose to compute
In order to obtain the results the computer must work several days. So we must be careful to avoid error. There are facts to verity the results.
At firstly, we compute the dimension of weight space, then by Sternberg tensor formula and Weyl formula we obtain the decomposition pattern of
1). Symmetry of dimension of weight space. Checking the results the two equations are satisfied:
2). Symmetry of composition factors. From the
3). Positivity of multiplicity of composition factors. All the multiplicity of composition factors we obtained are nonnegative.
4). Linkage principle is hold. If the multiplicity of composition factors
From the representation theory of algebraic groups, all the above results should be hold, so the computational data is compatible with the theory.
Theorem 2 When
Then
Remark: The table should be read as following. We list the weights in the first collum and write the multiplicity of composition factors as the others elements of tables. For example, from the third row in
Weight | Multiplicity of composition factors of irreducible module in Weyl module |
---|---|
00000 1 10001 1 1 00200 0 1 1 01002 0 1 0 1 20010 0 1 0 0 1 00111 1 1 1 1 0 1 11100 1 1 1 0 1 0 1 00030 1 0 0 0 0 1 0 1 00103 0 0 0 1 0 1 0 0 1 03000 1 0 0 0 0 0 1 0 0 1 30100 0 0 0 0 1 0 1 0 0 0 1 11011 2 2 1 1 1 1 1 0 0 0 0 1 11003 2 1 0 1 0 1 0 0 1 0 0 1 1 30011 2 1 0 0 1 0 1 0 0 0 1 1 0 1 00014 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 41000 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 30003 3 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 02020 2 0 1 0 0 1 1 1 0 1 0 1 0 0 0 0 0 1 10112 3 1 1 1 0 2 1 0 1 0 0 1 1 0 0 0 0 0 1 21101 3 1 1 0 1 1 2 0 0 0 1 1 0 1 0 0 0 0 0 1 10031 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 13001 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 00400 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 02004 2 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 40020 2 1 1 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 10023 2 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 32001 2 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 01121 2 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 12110 2 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 20202 6 1 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 00311 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 11300 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 01113 3 1 1 0 1 1 2 1 1 1 0 1 2 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 31110 3 1 1 1 0 2 1 1 0 1 1 1 0 2 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 40004 3 2 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 00303 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 30300 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 01032 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 23010 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 11211 6 0 2 0 0 2 2 1 0 1 0 1 1 1 0 0 0 2 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 11203 8 1 2 0 1 1 3 0 0 2 1 1 2 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 1 1 30211 8 1 2 1 0 3 1 2 1 0 0 1 1 2 0 0 1 1 1 1 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 30203 1 0 2 4 0 0 2 2 1 1 1 1 1 2 2 1 1 2 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 21212 1 0 3 7 1 1 3 3 3 1 3 1 1 2 2 1 1 1 2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 2 1 1 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module | |
---|---|---|
00001 1 12000 1 1 31000 0 1 1 00120 1 0 0 1 11020 2 1 0 1 1 00104 0 0 0 1 0 1 30020 2 1 1 0 1 0 1 22001 1 1 1 0 0 0 0 1 11004 2 0 0 1 1 1 0 0 1 10121 1 0 0 1 1 0 0 0 0 1 21110 2 1 1 1 1 0 1 1 0 0 1 30004 3 0 0 0 1 0 1 0 1 0 0 1 20300 0 0 0 1 0 0 0 0 0 0 1 0 1 10113 2 1 0 1 1 1 0 0 1 1 0 0 0 1 13010 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 10032 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 20211 2 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 01122 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 20203 4 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 00312 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 11212 4 2 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 | 10002 1 20100 0 1 00201 1 0 1 20011 1 1 0 1 20003 1 0 0 1 1 11101 1 1 1 1 0 1 03001 0 0 0 0 0 1 1 30101 0 1 0 1 0 1 0 1 02110 0 0 1 1 0 1 1 0 1 10202 1 0 1 1 1 1 0 0 0 1 41001 0 0 1 0 0 1 1 1 0 0 1 01300 0 0 1 0 0 0 0 0 1 0 0 1 10040 0 0 0 0 0 0 0 0 0 1 0 0 1 40110 1 0 1 1 0 1 1 1 1 0 1 0 0 1 01211 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 01130 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 01203 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 1 00320 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 01041 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 11220 0 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 30220 1 0 2 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 21221 2 1 3 1 2 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 | |
Weight | Multiplicity of composition factors of irreducible module in Weyl module |
---|---|
10210 1 02221 1 1 | 10010 1 01100 1 1 01011 1 1 1 01003 0 0 1 1 50000 0 1 0 0 1 00112 0 1 1 1 0 1 00031 0 0 0 0 0 1 1 11012 1 1 1 1 0 1 0 1 000230 1 0 1 0 1 1 0 1 30012 1 0 0 0 0 0 0 1 0 1 02021 0 1 0 0 0 1 1 1 0 0 1 21102 1 1 0 0 0 1 0 1 0 1 0 1 02013 1 1 0 1 0 1 1 1 1 0 1 0 1 12200 0 1 0 0 0 0 0 0 0 0 0 0 0 1 13002 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 40021 1 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 31200 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 32002 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 12111 0 2 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 0 1 40013 2 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 23100 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 12103 1 2 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 31111 1 3 1 1 1 2 1 1 0 2 1 1 0 1 1 1 1 1 1 0 0 0 1 31103 2 4 0 1 1 2 1 1 1 2 1 1 1 0 1 1 0 1 1 1 0 1 1 1 23011 0 2 0 0 1 1 0 0 0 1 0 1 0 1 2 0 1 1 1 0 1 0 1 0 1 23003 0 3 0 0 1 1 0 0 0 1 0 1 0 0 2 0 0 1 1 0 0 1 1 1 1 1 22112 3 8 1 1 3 2 1 1 1 2 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1 |
21021 1 21013 1 1 12022 1 1 1 | |
02102 1 40102 1 1 22120 1 1 1 | |
22010 1 10122 0 1 20212 1 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module |
---|---|
10012 1 01102 1 1 50002 0 1 1 02201 0 1 0 1 40201 1 1 1 1 1 12210 0 1 0 1 0 1 24001 0 0 0 1 1 0 1 31210 1 2 0 1 1 1 0 1 23110 0 2 1 1 1 1 1 1 1 22300 0 1 0 0 0 1 0 1 1 1 22211 1 2 1 1 1 1 1 1 1 1 1 | 10100 1 10011 1 1 10003 0 1 1 01101 1 1 0 1 00202 0 1 1 1 1 20012 1 1 1 0 0 1 50001 0 0 0 1 0 0 1 00040 0 0 0 0 1 0 0 1 11102 1 1 1 1 1 1 0 0 1 02200 0 0 0 1 0 0 0 0 0 1 03002 0 0 0 0 0 0 0 0 1 0 1 30102 1 0 0 0 0 1 0 0 1 0 0 1 02111 0 0 0 1 1 1 0 0 1 1 1 0 1 40200 0 1 0 1 0 0 1 0 0 1 0 0 0 1 41002 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 02030 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 02103 1 0 1 0 1 1 0 0 1 0 1 0 1 0 0 0 1 40111 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 24000 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 40030 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 40103 2 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 12120 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 1 31120 1 1 2 2 2 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 23020 0 0 0 2 1 0 1 0 1 1 2 1 0 1 1 0 0 0 1 0 0 1 1 1 22121 3 2 3 3 2 1 1 1 1 1 2 1 2 1 1 1 1 1 0 1 1 1 1 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module | |||
---|---|---|---|---|
12010 1 10212 1 1 10131 0 1 1 20221 1 1 1 1 | 02101 1 01202 1 1 01040 0 1 1 21220 1 1 1 1 | 01012 1 12201 0 1 31201 1 1 1 23101 0 1 1 1 22202 1 1 1 1 1 | 20101 1 01220 0 1 01204 1 1 1 01042 0 1 1 1 21222 1 1 1 1 1 | 20002 1 10201 1 1 01210 0 1 1 02220 0 1 1 1 12221 1 1 0 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module |
---|---|
00002 1 21000 0 1 00210 1 0 1 20020 1 1 0 1 12001 1 1 0 0 1 31001 0 1 0 0 1 1 11110 1 1 1 1 1 0 1 20004 1 0 0 1 0 0 0 1 10300 0 0 1 0 0 0 1 0 1 03010 0 0 0 0 1 0 1 0 0 1 30110 1 1 0 1 1 1 1 0 0 0 1 10211 1 0 1 1 1 0 1 0 1 0 0 1 10130 0 0 0 1 0 0 0 0 0 0 0 1 1 10203 1 1 0 1 1 0 0 1 0 0 0 1 0 1 10041 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 01212 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 20220 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 01131 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 00321 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 11221 2 1 1 1 1 0 1 1 2 1 1 1 1 1 1 1 1 1 1 1 | 00010 1 02000 1 1 40000 0 1 1 01020 1 1 0 1 01004 0 0 0 1 1 00121 0 0 0 1 0 1 00113 0 1 0 1 1 1 1 11021 1 1 0 1 0 1 0 1 00032 0 0 0 0 0 1 1 0 1 11013 2 1 0 1 1 1 1 1 0 1 30021 2 1 1 0 0 0 0 1 0 0 1 21200 0 1 1 1 0 0 0 0 0 0 0 1 22002 1 1 1 0 0 0 0 0 0 0 0 0 1 30013 3 0 0 0 0 0 0 1 0 1 1 0 0 1 13100 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 21111 2 1 1 1 0 1 0 1 0 0 1 1 1 0 0 1 02022 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 21103 4 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 13011 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 13003 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 12112 4 3 3 1 0 1 1 1 0 1 1 1 1 1 1 2 1 1 1 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module |
---|---|
00100 1 00011 1 1 11000 1 0 1 00003 0 1 0 1 30000 0 0 1 0 1 10020 1 1 1 0 0 1 02001 1 1 1 0 0 0 1 01110 1 1 1 0 0 1 1 1 10004 0 1 0 1 0 1 0 0 1 40001 0 0 1 0 1 0 1 0 0 1 00300 0 0 0 0 0 0 0 1 0 0 1 00211 0 1 0 1 0 1 1 1 0 0 1 1 11200 0 0 1 0 1 1 1 1 0 0 1 0 1 20021 1 1 1 1 1 1 0 0 0 0 0 0 0 1 12002 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 00130 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 00203 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 03100 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 30200 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 20013 1 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 31002 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 11111 1 1 1 1 1 2 2 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 11030 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 03011 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 11103 2 1 1 2 1 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 30111 2 1 1 1 2 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 1 00041 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 14000 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 30030 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 03003 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 30103 3 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 02112 1 0 1 1 3 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 2 0 1 1 0 0 0 0 1 0 1 21120 1 1 0 3 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 2 1 0 0 1 0 0 1 0 0 0 1 02031 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1 13020 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 12121 3 2 2 4 4 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 0 0 1 1 1 1 1 1 1 1 |
Weight | Multiplicity of composition factors of irreducible module in Weyl module | ||
---|---|---|---|
00122 1 22100 0 1 11022 1 0 1 22011 0 1 0 1 30022 0 0 1 0 1 22003 0 0 0 1 0 1 21112 1 1 1 1 1 1 1 | 01010 1 21012 1 1 12021 0 1 1 12013 1 1 1 1 31021 1 1 1 0 1 31013 2 1 1 1 1 1 22022 3 1 2 1 1 1 1 | 10101 1 20102 1 1 02120 0 1 1 02104 1 1 1 1 40120 1 1 1 0 1 40104 2 1 1 1 1 1 22122 3 1 2 1 1 1 1 | 00022 1 02012 1 1 40012 0 1 1 12102 0 1 0 1 31102 1 1 1 1 1 22200 0 0 0 0 0 1 23002 0 0 0 1 1 0 1 22111 1 1 1 1 1 1 1 1 |
According to the symmetry of
So we also have
We thank the Editor and the referee for their comments. This work was supported by the Natural Science Fund of Hohai University (2084/409277,2084/407188) and the Fundamental Research Funds for the Central Universities 2009B26914 and 2010B09714. The authors wishes to thank Prof. Ye Jiachen for his helpful advice.