On representations of non-semisimple specialized Hecke algebras
type: Item
instance of: scholarly article
title: On representations of non-semisimple specialized Hecke algebras
zbMATH Open document ID: 0791.20043
DOI: 10.1016/0021-8693(92)90017-G
author: Katsuhiro Uno
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: Let \((\mathcal{W,S})\) be a finite Coxeter system (see, for example, \textit{N. Bourbaki} [Groupes et algèbres de Lie, Chap. IV-VI (1968; Zbl 0186.330)]), \(q\) an indeterminate and \(H_ q({\mathcal W})\) the associated Hecke algebra. If \(\alpha\) is a complex number then \(H_ \alpha ({\mathcal W})\) is called a specialized Hecke algebra. \textit{A. Gyoja} and the author [J. Math. Soc. Japan 41, 75-79 (1989; Zbl 0664.20023)] proved that \(H_ \alpha({\mathcal W})\) is semisimple if and only if \(\alpha \neq 0\) and \(\alpha\) is not a root of the Poincaré polynomial \(P_{\mathcal W}(q)\) of \((\mathcal{W,S})\). The author conjectures that the algebra \(H_ \alpha({\mathcal W})\) is of finite representation type if and only if \(\alpha\) is a simple root of \(P_{\mathcal W}(q)\). The paper under review contains a proof of this conjecture for the particular case where \((\mathcal{W,S})\) is of type \(A\) or of rank 2.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 63612
zbMATH Keywords: finite Coxeter systems
zbMATH Keywords: specialized Hecke algebras
zbMATH Keywords: Poincaré polynomials
zbMATH Keywords: finite representation type
zbMATH Keywords: simple roots
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 63612