1. Introduction
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
’s with
for
being of type
,
,
,
and
in characteristic 2 and even in larger primes in [1] . Dowd and Sin gave all characters of
’s with
for all groups of rank less than or equal to 4 in characteristic 2 in [2] . They got their results by using the standard Gilkey-Seitz algorithm and computer. L. Scott et al. computes the characters for
when
,
by computing the maximal submodule in a baby Verma module [3] . Anders Buch and Niels Lauritzen also obtain this result for
when
with Jantzen’s sum formula [4] .
An element
for each irreducible module
with
was defined in [[5] , ![]()
39.1, p. 304] and [[6] , p. 239]. This element could be used in constructing a certain basis for
, computing
, and determining
. In this way, Xu and Ye, Ye and Zhou determined all irreducible characters for the special linear groups
,
and
, the special orthogonal group
and the symplectic group
over an algebraically closed field
of characteristic 2 in [7] [8] and for the special orthogonal group
and the symplectic group
over an algebraically closed field
of characteristic 3 in [9] [10] . However, it needs so much time to compute the irreducible characters for other groups. In the present note, we shall work out all irreducible characters for the simple algebraic groups of type
over an algebraically closed field
of characteristic 3 with modified algorithm to obtain faster speed. We shall freely use the notations in [9] [11] without further comments.
2. Preliminaries
Let
be the simple algebraic group of type
over an algebraically closed field
of characteristic 3. Take a Borel subgroup
and a maximal torus
of
with
. Let
be the character group of
, which is also called the weight lattice of
with respect to
. Let
be the root system associated to
, and choose a positive root system
in such a way that
corresponds to
. Let
![]()
be the set of simple roots of
such that
![]()
Let
be the fundamental weights of
such that
, the Kronecker delta, and denote by
the weight
with
, the integer ring. Then the dominant weight set is as follows:
![]()
Let
be the Weyl group and let
be the affine Weyl group of
. It is well-known that for
,
is the induced
-module from the 1-dimensional
-module
which contains a unique irreducible
-submodule
of the highest weight
. In this way,
parameterizes the finite-dimensional irreducible
-modules. We set
and
for all
. Moreover,
is given by the Weyl character formula, and for
, we have
![]()
For
, we have
![]()
Let
be the
-th Frobenius morphism of
with
the scheme-theoretic kernel of
. Let
be the Frobenius twist for any
-module
. It is well-known that
is trivial regarding as a
- module. Moreover, any
-module
has such a form if the action of
on
is trivial. Let
![]()
Then the irreducible
-modules
’s with
remain irreducible regarded as the
-modules. On the other hand, any irreducible
-module is isomorphic to exactly one of them.
For
, we have the unique decomposition
![]()
Then the Steinberg tensor product theorem tells us that
![]()
Therefore we can determine all the characters
with
by using the Steinberg tensor product theorem, provided that all the characters
with
are known.
Recall the strong linkage principle in [12] . We define a strong linkage relation
in
if
occurs as a composition factor in
. Then
is irreducible when
is a minimal weight in
with respect to the partial ordering determined by the strong linkage relations.
Let
be the simple Lie algebra over
which has the same type as
, and
the universal enveloping algebra of
. Let
,
be a Chevalley basis of
. We also denote
by
, respectively, where
The Kostant
-form
of
is the
-subalgebra of
generated by the elements
,
for
and
. Set
![]()
Then
for
,
,
. Define
and call
the hyperal-
gebra over
associated to
. Let
be the positive part, negative part, zero part of
, respec-
tively. They are generated by
,
and
, respectively. By abuse of notations, the images in
of
,
,
, etc. will be denoted by the same notations, respectively. The algebra
is a Hopf alge-
bra, and
has a triangular decomposition
. Given a positive integer
, let
be the sub-
algebra of
generated by the elements
,
,
for
,
and
. In
particular,
is precisely the restricted enveloping algebra of
. Denote by
the positive part, negative part, zero part of
, respectively. Then we have also a triangular decomposition
. Given an ordering in
, it is known that the PBW-type bases for
resp. for
have the form of
![]()
with
resp. with
.
Let
. We set
for
, here each element
is also viewed as a certain set of simple roots. Following [5] [6] , we define an elements
in
by
![]()
As a special case of [[5] , Theorems 6.5 and 6.7], we have
Theorem 1 Assume that
is a simple Lie algebra of the simple algebraic group of type
over an algebraically closed field
of characteristic 3. Let
.
(i) The element
lies in
.
(ii) Let
be the left ideal of
generated by the elements ![]()
and the elements
with
. Then
(Note that
has a
-module structure, which is irreducible).
(iii) As a
-module,
is isomorphic to
.
By abuse of notations, the images in
of
and
will be denoted by the same notations. We shall use this theorem to computer the multiplicities of the weight spaces for all the dominant weight of
, to compute
, and to determine
in this note, when
is the simple algebraic group of type
.
3. Characters of Irreducible Modules of ![]()
From now on we shall assume that
. Denote by
the dual module of
, then we have by the duality that
, and
. Furthermore, the elements
satisfy the following commutator relations:
![]()
Now we can obtain our main theorems. Let
be the sum of weights of the W-orbit of ![]()
for all
. It is well-known that
,
and ![]()
form bases of
, the W-invariant subring of
, respectively. According to the Weyl character formula and the Freudenthal multiplicity formula, we get a change of basis matrix
from
to
, which is a triangular matrix with 1 on its diagonal, i.e.
![]()
with
(cf. [10] ). Based on our computation, we get another change of basis matrix
![]()
from
to
, which is also a triangular matrix with 1 on its diagonal.
Let us mention our computation of
more detailed. First of all, we compute
for any
. It is well known that for each dominant weight
of
,
can be expressed in terms of sum of positive roots, and there exist many ways to do so. Each way corresponds to an element
in
. Then we compute various
. Note that each
can be written as a linear combination of the basis elements of
with non-negative integer coefficients, and the typical images of all non-zero
’s generate the weight space
of the irreducible submodule
of
. Therefore, we can easily determine the dimension of
, provided that we compute the rank of the set of all these non-zero
’s. It can be reduced to compute the rank of a corresponding matrix. Finally, we obtain the formal character of
, which can be written as a linear combination of
’s with non-negative integer coefficients. That is
![]()
with
. In this way, we get the second matrix
.
For example, we assume that
is the simple algebraic group of type
and
.
It is easy to see that
![]()
For
, we have
First we compute each of the set
. Then we compute the rank of the set
, which is equal to 2. So we have
. For
, we have
We compute each of the set
![]()
and then we compute the rank of the set
, which is equal to 13. So we have
. By this methods, we can calculate all multiplicity
Finally, we obtain the formal character of irreducible module ![]()
When
lies in
but not in
, we can also compute the formal character
by using the Steinberg tensor product theorem. For
, we have the unique decomposition
![]()
Then the Steinberg tensor product theorem tells us that
![]()
Therefore, we can determine all characters
with
, provided that all characters
with
are known. For example, when
, we have
![]()
Therefore, from the two matrices
, we can easily get the third change of basis matrix
from
to
, which is still a triangular matrix with 1 on its diagonal. The matrix
gives the decomposition patterns of various
with
.
We list the matrix
in the attached tables. In all these tables, the left column indicates
’s. For two weight
, the number
in tables is just the multiplicity of composition factors
.
4. Faster Algorithm
In paper [9] [10] , we compute the multiplicity
one by one for a fixed weight
However, noticing that some information computing
may be useful to compute
for
So we compute all possible
such that
spanning to the whole
firstly. Then we compute
in some ordering: if
then we first obtain
save this result and compute
instead of computing
directly. In fact we only need compute
for some positive root
and
in one step.
For example, suppose to compute
we can compute
at the first step, and then compute
In this way, we can avoid much repeated work.
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
At last checking all the data we find that
1). Symmetry of dimension of weight space. Checking the results the two equations are satisfied:
![]()
2). Symmetry of composition factors. From the
decomposition patterns, the following equations are hold:
![]()
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
then we have ![]()
From the representation theory of algebraic groups, all the above results should be hold, so the computational data is compatible with the theory.
5. Main Results
Theorem 2 When
, let
![]()
Then
is an irreducible
-module for all
and the decomposition patterns of
for all
are listed in Tables 1-8.
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 Table 1, we obtain 00200 0 1 1, this mean
![]()
![]()
Table 1. The linkage class (00000).
![]()
Table 2. The linkage class (00001), (10002).
![]()
Table 3. The linkage class (10210), (21021), (02102), (22010), (10010).
![]()
Table 4. The linkage class (10012), (10100).
![]()
Table 5. The linkage class (12010), (02101), (01012), (20101), (20002).
![]()
Table 6. The linkage class (00002), (00010).
![]()
Table 7. The linkage class (00100).
![]()
Table 8. The linkage class (00122), (01010), (10101), (00022).
According to the symmetry of
we need not list all results. For example, we can obtain the decomposition pattern of
from Table 2:
![]()
So we also have
![]()
Acknowledgements
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.