Presenting Hecke endomorphism algebras by Hasse quivers with relations
Presenting Hecke endomorphism algebras by Hasse quivers with relations
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DOI：
10.1016/j.jpaa.2016.08.010
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发表时间：
2015-11
期刊：
影响因子：
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通讯作者：
J. Du;B. T. Jensen;Xiuping Su
中科院分区：
文献类型：
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作者：
J. Du;B. T. Jensen;Xiuping Su
Hecke自同态代数是与对称群相关的Q-Schur代数到Coxeter群的自然推广。对于Weyl群，B.Parshire，L.Scott和第一作者[9]，[10]研究了这些代数的分层结构，以寻求对Lie型有限群表示的应用。本文研究了伴随于任意Coxeter群的Hecke自同态代数的表示问题。我们的方法是用带有关系的箭图来表示这种代数。如果R是Z[Q]在常数项为1的多项式上的局部化，则代数可以简单地由所谓的幂等、夹心和扩展辫子关系来定义。作为这一结果的应用，我们首先得到了Z上0-Hecke自同态代数的一个表示，然后通过寻找挠关系发展了一个表示Z[Q]上的Hecke自同态代数的算法。作为例子，我们确定了所有阶2群和对称群S 4所需的挠率关系。
A Hecke endomorphism algebra is a natural generalisation of the q-Schur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, B. Parshall, L. Scott and the first author [9],[10] investigated the stratification structure of these algebras in order to seek applications to representations of finite groups of Lie type. In this paper we investigate the presentation problem for Hecke endomorphism algebras associated with arbitrary Coxeter groups. Our approach is to present such algebras by quivers with relations. If R is the localisation of Z [q] at the polynomials with the constant term 1, the algebra can simply be defined by the so-called idempotent, sandwich and extended braid relations. As applications of this result, we first obtain a presentation of the 0-Hecke endomorphism algebra over Z and then develop an algorithm for presenting the Hecke endomorphism algebras over Z [q] by finding torsion relations. As examples, we determine the torsion relations required for all rank 2 groups and the symmetric group S 4.