1. Introduction
Computing the Cartan invariant matrix 
for a finite group of Lie type is an important research subject in the representation theory. One has made many great efforts for this.
Firstly, let finite group be of type
. 
for 
and 
are treated combinatorially by Alperin [1] [2] and Upadhyaya [3] that almost all Cartan invariants are powers of 2 but some exceptional ones for the form
. Benson, Martin [4] and Humphreys [5] exhibit 
for 
and
. 
for 
is given by Humphreys [6] when 
and by Ye [7] when 
. Zaslawsky [8] computes 
for 
when 
and for 
when
. 
for 
and 
is computed by Benson [9] and Du [10] . 
for 
has been computed independently by Jantzen (unpublished work) and Ye [11] .
Secondly, let finite group be of type 
and
. Humphreys [6] and Thackray [12] work out independently 
for 
. Similar computations for 
are summarized in [13] but 
is not exhibited. Ye [14] , Ye and Zhou [15] , Y. Cheng (unpublished work), Liu and Ye [16] , Hu and Xu [17] compute 
for 
, 
, 
, 
, 
, respectively.
At last, let finite group be of type
. 
for 
has been computed by Mertens [18] when 
and Hu, Ye [19] when 
. In the present note, we shall compute 
for the finite group 
of type 
over a field 
with 
elements. Some computations involved in this paper were done by using a computer and the MATLAB software. We shall freely use the notations in [15] without further comments.
2. Preliminaries
Let 
be a simply-connected semisimple algebraic group of type 
over an algebraically closed field 
with characteristic
. Take a maximal torus 
of 
such that 
is the weight lattice of 
with respect to
. Let 
be the root system associated to 
with the two simple roots
, where 
is the long simple root, then 
is the set of positive roots, and 
be root lattice in
. Let 
be the set of dominant weights with the corresponding fundamental weights 
satisfying
, and denote by 
the weight 
with
. Then
It is well known that 
with 
is the Weyl module of the highest weight 
with the unique simple quotient module
. In this way, 
parameterizes the finite dimensional simple 
-modules.
Let 
denote the scheme-theoretic kernel of the 
-th Frobenius morphism 
of 
and 
the Frobenius twist for any 
-module
. It is well known that 
is trivial as a 
-module. Moreover, any 
-module that becomes trivial upon restriction to 
is of this form. Let
be the set of restricted dominant weights, then the simple 
-modules
’s with 
remain simple regarded as the 
-modules. On the other hand, any simple 
- module is isomorphic to exactly one of them. Denote by 
the simple 
-modules with
, whose projective cover is
. Then 
is an index set of isomorphic classes of simple 
-modules. Let 
be the finite subgroup consisting of all fixed points of 
in
, which is called the finite group of type
. The following facts are well known. For
, the restriction of the simple 
-module 
to
, denoted by
, remains simple. Furthermore, any simple 
-module is isomorphic to exactly one 
with
. We denote by 
the projective indecomposable
-module (or 
-PIM, for short), which has 
as its top and bottom composition factors.
Moreover, the restriction 
is also a projective 
-module and it is decomposed into a direct sum of 
-PIM’s such that 
occurs exactly once. 
is also an index set of isomorphic classes of simple 
-modules 
and of projective indecomposable 
-modules
.
By the definition, the Cartan invariant 
of 
is the multiplicity of the simple
-module 
occurring as a 
-composition factor of the projective indecomposable
-module
, i.e.
. Symmetric matrix 
is called the Cartan invariant matrix, it is of 
order.
From now on, we assume
. For any
, write
,
,
. And for any
, write
,
.
3. G-Composition Factors of 
Let 
be the affine Weyl group of
, which is generated by all affine reflections 
with
, where 
sends 
to
. Dot action 
of 
on 
is defined as follows: 
for 
and
. For this dot action, the origin of 
is placed at
, and 
is partitioned into alcoves, whose closures are fundamental domains for the action of 
on the euclidean space
. In case of this paper, 
lie in a parallelogram with lowest point 
and highest point
, which is a union of 12 restricted alcoves (see Figure 1). Following [20] , any 
-translate of this parallelogram is called a “box”; highest point of every box is a special point (intersection of all possible types of affine reflecting hyperplanes). We number some of the dominant alcoves as in Figure 2 for easy reference. We say that an alcove is of type 1 if it is a 
-translate of the alcove marked 1, and so on, for the various restricted alcoves marked 1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 15, 16.
consists of highest weight
, weights inside 
alcoves and weights on 
walls. Let 
be any positive integer. We denote by 
the set of all weights 
inside the alcove marked
. Then weights inside 12 restricted alcoves can be represented as follows:
;
;
;
;
;
;
;
;
;
;
;
And, we denote by
, 
, 
the sets of all weights on the short right-angled edge, the long right-angled
edge, the incline edge of the right triangle which circles the above marked
, respectively, then we can describe weights on 
walls. For example, 
, 
, 
, and so on. It is well known that Weyl group 
of 
is a Coxeter group generated by 
and
. Write
, then
, where
. And, the dot action of 
on 
induces the dot action of 
on
.
, saying that 
is linked in 
means that there exists 
such that
, and write
.
Recall some results of Jantzen [21] or Humphreys [22] on the generic decomposition patterns of Weyl modules, limiting ourselves to weights which lie in the lowest 
-alcove (an alcove for the affine Weyl group relative to
). When a dominant weight 
lies inside an alcove sufficiently far from the walls of the dominant Weyl chamber, the pattern of 
-composition factors of 
depends only on the type of alcove in which 
lies. The corresponding “generic decomposition pattern” consists of the alcoves which contain reflected weights 
for which 
occurs as a 
-composition factor of
; each such alcove is labelled with the multiplicity of 
as a 
-composition factor. For type 
there are 12 type of alcoves, corresponding to the 12 alcoves in the restricted box; hence there are 12 generic patterns. In particular, all patterns involve the same number of alcoves and the same distribution of multiplicities. For type 
the total number of composition factors is 119. Figure 3 shows generic decomposition pattern of 
with 
lying inside alcove of type 1. Here the bold alcove marked 
means the alcove in which the highest weight of 
lies, and 
appears once as 
-composition factor of
, other alcoves marked digits 1 to 4 refer to those alcoves in which the remaining 
-composition factors
’s with the corresponding multiplicities 1 to 4 of 
except 
lie.
Now we determine the 
-composition factors of 
with
. If 
lies inside an alcove, we consider each alcove in the generic pattern which lie outside the dominant chamber. Find the special point at the top of the unique box in which that alcove lies. If that point lies on a reflecting hyperplane through
, discard the alcove. Otherwise there is a unique element 
taking that point to the special point at the top of a box in the dominant chamber. Find the alcove in this box corresponding to the given alcove, and attach to it the multiplicity in the given alcove, with a sign equal to det
. After carrying out this process for all alcoves, and cancelling multiplicities if necessary, the end result is the pattern of 
-composition factors of 
with
. Figure 4 illustrates this algorithm when 
is inside alcove 25 (marked
) which is of type 11. The two long bold lines indicate the walls of the dominant chamber, i.e., 
-wall and 
-wall. The bold alcoves marked digits 2, 1, 1, 1, 1, 1, 1 to the right of the 
-wall correspond to the dominant alcoves marked digits 4, 3, 3, 3, 2, 1, 2, respectively. A single reflection 
is involved here, so there are some multiplicity cancellations of alcoves marked digits 4, 3, 3, 3, 2, 1, 2 from the figure. All other alcoves to the right of the 
- wall disappear, since the special points at the tops of their boxes lie on reflecting hyperplanes through
. Similarly, The bold alcoves marked digits 1, 1, 1, 1 to the left of the 
-wall correspond to the dominant alcoves marked digits 4, 3, 3, 3, respectively. A single reflection 
is involved here, so there are some multiplicity cancellations of alcoves marked digits 4, 3, 3, 3 from the figure. All other alcoves to the left of the 
-wall also disappear, since the special points at the tops of their boxes lie also on reflecting hyperplanes through
. So the end result is that 
will have 27 
-composition factors, corresponding to 
inside alcove
Figure 3. General decomposition pattern for alcove of type 1.
Figure 4. Patterns of G-composition factors of alcove 25.
25 and the reflected weights inside alcoves 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12(twice), 13, 14(twice), 15, 16, 17(twice), 18, 19, 20, 21, 22, 23, 24.
For a weight 
not lying inside an alcove, the 
-composition factors of 
are obtained by using Jantzen’s translation principle (cf. [23] ). Find the alcove in whose “upper closure” 
lies and compute the pattern as above for an interior weight of this alcove. Then translate all weights involved to the type of wall in which 
lies; only those in upper closures of alcoves survive to give composition factors of
. For example, when
, 
will have 13 
-composition factors, corresponding to 
on wall 
and the reflected weights on walls
, 
, 
, 
, 
(twice), 
, 
, 
(twice), 
,
.
In particular, we have the following Proposition.
Proposition 3.1 The decomposition patterns of 
-composition factors
’s of 
with 
are listed as follows (Table 1). Here
’s are the reflected weights of 
under
.
4. Weyl Filtrations of 
Module 
Jantzen [24] shows that when
, principal indecomposable 
-module 
admits a 
-module filtration with Weyl 
-modules as subquotients so called Weyl filtration, and the times of Weyl 
-module 
appearing as a subquotient in Weyl filtration of 
equals to the multiplicity of the simple 
-module 
occurring as a 
-composition factor of Weyl 
-module
, i.e.,
. In the case of this paper,
. By a lots of concrete computations, we find that when
, there are at most 80 dominant alcoves corresponding to the highest weights of Weyl 
-modules occurring as subquotients in Weyl filtration of 
with
, these 80 dominant alcoves are figured in Figure 2. So we can get Weyl 
-modules
’s occurring as subquotients in Weyl filtration of 
with 
from decomposition patterns described in
. For 
lying inside restricted dominant alcoves, we have the following Proposition 4.1.
Proposition 4.1 Assume that digit 1 denotes a typical weight inside the alcove 1, digits 
denote the reflected weights inside the alcoves 
under W13, then we have the following Table 2. Here the first column denotes the principal indecomposable 
-modules
, the second column denotes the corresponding Weyl Gmodules
’s, and n2 means that 
occurs twice.
Similarly, we can obtain Weyl 
-modules
’s occurring as subquotients in Weyl filtration of 
with 
on walls, i.e., upper closures of restricted dominant alcoves.
For example, 
lying inside alcove 16, we can get from the last row of Table 2 that
and
, so we have
By Proposition 4.1, we get Weyl 
-module
’s decompositions of 
with
, and by the methods given in
, we obtain 
-composition factors of Weyl 
-module
’s, so we can get all
- composition factors of 
with
. For example
Table 1. G-composition factors L(μ)’s of
.
5. 
-Composition Factors of 
Let 
be the set of all weights of 
for
, and 
for
, i.e., the multiplicity of 
in
. Some values of 
are determined in Table 3, where the first row contains the weights 
each of which labels the corresponding column, and the first column contains the wights 
each of which labels the corresponding row. The number lying in the intersection of the 
-row and 
-column is just
.
For
, let
, then we have
, where 
if 
lie on wall of 
chamber or 
if 
do not lie on wall of 
chamber and
. By using Table 3, we have
, for all
.
For
, by 
and Proposition 4.1, we have
, where
, 
is the weight linked to
; 
and
. As 
-module, we
have
, so can determine 
as follows. In case of this paper,
. If
, then
, i.e., 
, we get all 
-composition factors of
; if
, which 
is maximal one in weights less than 
linked to 
in
, then we have
. But weights less than 
linked to 
in 
have at most 12, and the Weyl module corresponding to the smallest one is simple. Repeatedly, we have
, where
.
If
, then
; if
, then
, 
or
, so
, return to the above case. In the end, we have
, where
. so determine all 
-composition factors
’s 
of
.
For example, we have the following expression:
6. 
-Decomposition of 
For any
, restriction of 
-module 
on 
is projective 
-module, and can be decomposed into the direct sum of the principal indecomposable 
-module
, Chastkofsky [25] and Jantzen [26] show the decomposition formula, i.e.,
Using this formula and the knowledge of decomposition of tensor products of two simple 
-modules in the above section, and by a series of complicated computations, we have the following Proposition 6.1.
Proposition 6.1 The character 
of 
for 
can be expressed as a sum of character 
of 
with 
as in the following Table 4.
By the Proposition 6.1 and the above
, we can get all the Cartan invariants of 
. For example, since
, from the above 4.3, we have
, and so on.
7. Main Results
Let us arrange the row indices from left to right and the column indices from top to bottom in the following order: 
for 
taking
, successively;
for 
taking
, successively;
for 
taking
, successively;
for 
taking
, successively; (12,12). Then, we write the Cartan invariant matrix 
for the finite group 
of type 
as
. For convenience, we block 
into
, where 
with 
is square matrix of 
order, and
, 
with 
are 
matrices, and
, 
are square matrices of 
order. We list all matrices 
in Table 5, where the elements below diagonal are omitted for
, the elements above diagonal are omitted for
.
By MATLAB soft, we have
, which is also known by a general result in the representation theory of finite groups on the determinant of the Cartan invariant matrix.
Acknowledgements
The first version of this paper was done during the first named author visited the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy from 1, July, 2009 to 31, August, 2009. He would like to thank the ICTP for the financial support during his stay. He would also like to thank the Commission on Development and Exchanges of the International Mathematical Union for the financial support of the travel expenses in connection with his visiting to the ICTP.
NOTES
*Supported by the Science Foundation for the Excellent Youth Scholars of Henan Province (Grant No. [2005]461).
#Corresponding author.