On representations of non-semisimple specialized Hecke algebras
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.
scientific article; zbMATH DE number 63612
scientific article