Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras
Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras
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DOI：
10.1112/jlms/s2-44.3.420
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发表时间：
1991-12
影响因子：
1.2
通讯作者：
J. Du;B. Parshall;Jian-pan Wang
中科院分区：
文献类型：
--
作者：
J. Du;B. Parshall;Jian-pan Wang
设KBE是交换环K上的n阶自由模，r次对称群sr自然作用于张量空间K？r.经典地，Schur代数SF(n，r)与自同态代数Ends(K？r)一致，见JA Green[11].在一个重要的发展中，R.Dipper和G.James推广了Schur代数的概念，得到了#-Schur代数^q(n，r)[4，5]。因此，对于Hecke代数Jf=if9(Sr)在‘^-张量空间’V®r上的作用，Sfq(Nj)=end^K®1“).当q=1时，GB Fq(n，r)与上面定义的Schur代数Sf(n，r)重合.如[11]所示，SF(n，r)具有涉及一般线性群GL(？，K)的另一种描述。因此，Schur代数在这些还原代数群的表示理论中起着重要的作用。最近，关于^-Schur代数的量子群的一个类似的解释也被发现。在文[3]中，R.Dipper和S.Donkin定义了一个量子一般线性群GLQ(n，K)‘，并得到了关于这个群的代数SfQ(n，r)。独立地，帕尔希尔和
Let Kbe a free module of rank n over a commutative ring K. The symmetric group Sr of degree r acts naturally on the tensor space K® r. Classically, the Schur algebra Sf (n, r) identifies with the endomorphism algebra Ends (K® r); see JA Green [11]. In an important development, R. Dipper and G. James extend the notion of a Schur algebra to obtain the#-Schur algebras^ Q (n, r)[4, 5]. Thus, Sfq {nj)= End^ K® 1") for an action of the Hecke algebra Jf= if9 (Sr) on a'^-tensor space'V® r. When q= 1,£ fQ (n, r) coincides with the Schur algebra Sf (n, r) defined above. As shown in [11], Sf (n, r) has an alternative description involving the general linear group GL («, K). Therefore, Schur algebras play an important role in the representation theory of these reductive algebraic groups. Recently, a similar interpretation of^-Schur algebras in terms of quantum groups has been found. In [3], R. Dipper and S. Donkin define a certain quantum general linear group GLQ (n, K)'and obtain the algebras SfQ (n, r) in terms of this group. Independently, Parshall and