On endomorphism algebras arising from multiparameter Hecke algebras
type: Item
instance of: scholarly article
title: On endomorphism algebras arising from multiparameter Hecke algebras
zbMATH Open document ID: 0888.20012
author: Hebing Rui
published in: Journal of Algebra
publication date: 1998-06-07T00:00:00Z
review text: Let \(\{G(q)\}\) be a family of finite groups of Lie type having \((W,S)\) as irreducible Coxeter system and let \(H\) be the Hecke algebra corresponding to \(G(q)\) over \(A=\mathbb{Z}[q^{1/2},q^{-1/2}]\); the ring of Laurent polynomials in the indeterminate \(q^{1/2}\). For any \(I\subseteq S\), let \(x_I=\sum_{w\in W_I} T_\omega\), where \(W_I\) is the standard parabolic subgroup of \(W\) generated by \(I\). Let \({\mathcal A}=\text{End}_H(\oplus_I x_iH)\). When \(G(q)=\text{GL}(n,q)\) then \({\mathcal A}\) is known as a \(q\)-Schur algebra. In this paper the author considers the case when \(G=U_{2n}(q)\) or \(U_{2n+1}(q)\) and studies the representation theory of \({\mathcal A}\) in those cases by using Kazhdan-Lusztig theory. The main result (Theorem 2.7) is that the Kazhdan-Lusztig basis of \({\mathcal A}\) is a standard basis (see 2.1 for definition). He also describes the simple modules and standard modules in the category of left \({\mathcal A}\)-modules.
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1076277
zbMATH Keywords: \(q\)-Schur algebras
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: irreducible Coxeter systems
zbMATH Keywords: Hecke algebras
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: Kazhdan-Lusztig bases
zbMATH Keywords: simple modules
reviewed by: Ahmed A. Khammash
MaRDI profile type: Publication
full work available at URL: jabr.1997.7032
OpenAlex ID: W2023231741
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DOI: 10.1006/JABR.1997.7032
description: scientific article; zbMATH DE number 1076277