Semisimplicity of parabolic Hecke algebras
type: Item
instance of: scholarly article
title: Semisimplicity of parabolic Hecke algebras
zbMATH Open document ID: 0913.20005
author: Yasushi Gomi
published in: Journal of Algebra
publication date: 1999-06-08T00:00:00Z
review text: Let \((W,S)\) be a finite Coxeter system and let \(R:=\mathbb{Z}[r_s;\;s\in S]\) be the polynomial ring on the set \(\{r_s;\;s\in S\}\) of indeterminates chosen so that \(r_s=r_t\Leftrightarrow s\) and \(t\) are conjugate in \(W\) and \(q_s=r^2_s\). The Hecke algebra \(H(S,R)\) is an \(R\)-algebra with basis \(\{T_w;\;w\in W\}\) and multiplication given by \[ T_sT_w=\begin{cases} T_{sw}\quad &\text{if }\ell(sw)>\ell(w)\\ q_sT_{sw}+(q_s-1)T_w\quad &\text{if }\ell(sw)<\ell(w)\end{cases} \] for \(s\in S\); \(w\in W\), where \(\ell\) is the length function on \((W,S)\). The algebra \(H(S,R)\) is called Iwahori-Hecke algebra associated with \((W,S)\). For \(J\subseteq S\), the parabolic Hecke algebra \(H(S,J;R)\) is a subalgebra of \(H(S;R)\) defined as \(H(S,J;R):=\{a\in H(S;R):T_sa=aT_s=q_sa;\;\forall s\in J\}\). This paper determines the semisimplicity of the parabolic Hecke algebra \(H(S,J;R)\). It is an extension of an earlier paper by the author [J. Algebra 183, No. 2, 514-544 (1996; Zbl 0856.20009)] in which he studied the semisimplicity of parabolic Hecke algebras when they have only one parameter \(q\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1203916
zbMATH Keywords: finite Coxeter systems
zbMATH Keywords: Hecke algebras
zbMATH Keywords: length functions
zbMATH Keywords: Iwahori-Hecke algebras
zbMATH Keywords: parabolic Hecke algebras
zbMATH Keywords: semisimplicity
reviewed by: Ahmed A. Khammash
MaRDI profile type: Publication
full work available at URL: jabr.1997.7312
OpenAlex ID: W2095318976
cites work: The representations and generic degrees of the Hecke algeba of type H4.
cites work: On the Degrees and Rationality of Certain Characters of Finite Chevalley Groups
cites work: Q3682663
cites work: Hecke algebras and characters of parabolic type of finite groups with (B, N)-pairs
cites work: Q3922829
cites work: Q4724810
cites work: Projective simple modules of symmetric algebras and their specializations with applications to Hecke algebras
cites work: Structure constants of the Hecke algebras \({\mathcal H}(S,J)\) and applications
cites work: On the Steinberg characters of finite Chevalley groups
cites work: Modular representation theory over a ring of higher dimension with applications to Hecke algebras
cites work: On the semisimplicity of Hecke algebras
cites work: On a theorem of Benson and Curtis
cites work: Q4750022
cites work: Q3947818
cites work: Q3739243
cites work: Representation theory of Weyl group of type \(C_n\)
cites work: Irreducible projective modules of the Hecke algebras of a finite Coxeter group
DOI: 10.1006/JABR.1997.7312
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description: scientific article; zbMATH DE number 1203916