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 for has been computed independently by Jantzen (unpublished work) and Ye [11] .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 Ye [14] , Ye and Zhou [15] , Y. Cheng (unpublished work), Liu and Ye [16] , Hu and Xu [17] compute for , , , , , respectively.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 W_{13}, 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 n^{2} 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.