$q$-tensor space and $q$-Weyl modules
$q$-tensor space and $q$-Weyl modules
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DOI：
10.1090/s0002-9947-1991-1012527-1
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发表时间：
1991
影响因子：
1.3
通讯作者：
R. Dipper;G. James
中科院分区：
文献类型：
--
作者：
R. Dipper;G. James
We obtain the irreducible representations of the q-Schur algebra, motivated by the fact that these representations give all the irreducible representations of GLn(q) in the nondescribing characteristic. The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction. The theory of polynomial representations of general linear groups is equivalent to the representation theory of Schur algebras (see Green's book [5] and the bibliography therein). In [4], we defined q-analogues of Schur algebras. When q = 1, these are the usual Schur algebras, and when q is a prime power, representations of q-Schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case, including all irreducible representations of these groups and important information about decomposition numbers. It is natural to ask what features of the classical Schur algebras have qanalogues. In this paper, we define q-analogues of tensor space, of Weyl modules, and of weight spaces, thereby generalizing the main reslllts which appear in Green's book [5]. For example, we classify the irreducible modules for qSchur algebras, we determine bases for q-Weyl modules compatible with weight spaces, and we give results on composition multiplicities of the irreducible modules in q-Weyl modules. The proofs are largely self-contained, so by specializing q to 1, we recover the corresponding results in [5]. This paper, therefore, is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics. 1. THE q-SCHUR ALGEBRA Let r be a natural number, let R be an integral domain, and let q be a unit in R. We denote the symmetric group on r letters by er . The Hecke algebra Z is the R-free R-algebra with basis {Twlw E er} where the multiplication is determined by the following rule. If a = (i, i + 1) is a basic transposition in Received by the editors August 15, 1989. 1980 Mathematics Subject Classification ( 198 5 Revision) . Primary 1 6A64, 1 6A6 5 ; Secondary 20C30. This research was supported in part by NSF Grant No. DMS-8802290. The authors gratefully acknowledge support received from NATO Grant No. 0222/87. (r) 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page
