Automorphisms of generic Iwahori-Hecke algebras and integral group rings of finite Coxeter groups
type: Item
instance of: scholarly article
title: Automorphisms of generic Iwahori-Hecke algebras and integral group rings of finite Coxeter groups
zbMATH Open document ID: 0898.20004
author: Frauke Bleher
author: Meinolf Geck
author: Wolfgang Kimmerle
published in: Journal of Algebra
publication date: 1998-11-03T00:00:00Z
review text: Let \(G\) be a finite group and let \(Cl(G)\) be the set of conjugacy classes of \(G\). The group \(\Aut CT(G)\) of character table automorphisms is defined to be a bijection \(\tau\colon Cl(G)\to Cl(G)\) such that the induced map \(\tau\colon Z(\mathbb{Z} G)\to Z(\mathbb{Z} G)\) is a unitary ring automorphism which commutes with all power maps. The authors prove that if \((G,S)\) is a finite Coxeter system, then \(\Aut CT(G)\simeq\Aut(G)/\text{Inn}(G)\), the group of outer automorphisms of \(G\). Moreover, the authors prove that if \((G,S)\) is a finite Coxeter system and if \(H\) is a finite group such that \(\mathbb{Z} G\simeq\mathbb{Z} H\) as augmented algebras, then there is a unit \(u\) in \(\mathbb{Q} G\) such that \(uGu^{-1}=H\). The authors establish the structure of algebra automorphisms of any Hecke algebra. They prove that the automorphism group is a semidirect product of the central automorphisms by the direct product of the graph automorphisms and the group generated by the Alvis-Curtis duality. The article contains a wealth of information on automorphism groups of finite groups in general, of finite Coxeter groups and character table automorphisms. The heart of the proof of the above results is a detailed discussion of the standard reflection character of the Coxeter group. The authors prove by a lengthy case by case analysis that the reflection character is fixed by a character table automorphism if and only if it is induced by a graph automorphism. This reflection character takes the role of the trivial character in [\textit{F. Bleher}, Automorphismen von Gruppenringen und Blocktheorie (Dissertation Stuttgart 1996; Zbl 0858.20001)]\ where she used this character to establish analogous statements for certain blocks with cyclic defect groups. Work of Geck and Geck, Rouquier set up a theory similar to blocks with cyclic defect groups and using this and similar considerations as in the first author's dissertation, the authors establish the above stateme…
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 1104118
zbMATH Keywords: finite groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: character table automorphisms
zbMATH Keywords: finite Coxeter systems
zbMATH Keywords: groups of outer automorphisms
zbMATH Keywords: algebra automorphisms
zbMATH Keywords: Hecke algebras
zbMATH Keywords: automorphism groups
zbMATH Keywords: central automorphisms
zbMATH Keywords: graph automorphisms
zbMATH Keywords: finite Coxeter groups
reviewed by: Alexander Zimmermann
describes a project that uses: GAP
describes a project that uses: CHEVIE
MaRDI profile type: Publication
full work available at URL: jabr.1997.7118
OpenAlex ID: W2088266951
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DOI: 10.1006/JABR.1997.7118
description: scientific article; zbMATH DE number 1104118