The Glauberman character correspondence and perfect isometries for blocks of finite groups
type: Item
instance of: scholarly article
title: The Glauberman character correspondence and perfect isometries for blocks of finite groups
zbMATH Open document ID: 0936.20006
author: Atumi Watanabe
published in: Journal of Algebra
publication date: 2000-05-15T00:00:00Z
review text: Let \(G\) be a finite group and let \(S\) be a solvable finite group that acts on \(G\) and such that \((|G|,|S|)=1\). Let \(p\) be a prime and let \((K,O,F)\) be a \(p\)-modular system such that \(K\) is ``big enough'' for \(G\). For a central idempotent \(e\) of \(OG\), let \(R_K(G,e)\) denote the additive group of characters of \(G\) afforded by \(KGe\)-modules. Let \(\text{Irr}(G)\) denote the set of ordinary irreducible characters of \(G\) and let \(\text{Irr}_S(G)\) denote the set of \(S\)-invariant elements of \(\text{Irr}(G)\). As is well-known, in this case, the canonically defined Glauberman correspondence gives a bijection \(\pi(G,S)\colon\text{Irr}_S(G)\to\text{Irr}(C_G(S))\). The first two main results of the paper are: Theorem 1. Let \(B\) be an \(S\)-invariant \(p\)-block of \(G\) such that \(C_G(S)\) contains a defect group \(D\) of \(B\). Then \(\text{Irr}(B)\subseteq\text{Irr}_S(G)\) and there exists a unique block \(b\) of \(C_G(S)\) with defect group \(D\) such that \(\text{Irr}(b)=\pi(G,S)(\text{Irr}(B))\). Moreover there is a perfect isometry \(R\colon{\mathcal R}_K(G,B)\to{\mathcal R}_K(C_G(S),b)\) such that \(R(\chi)=\pm\pi(G,S)(\chi)\) for any \(\chi\in\text{Irr}(B)\). Theorem 2. In the situation of Theorem 1, the Brauer categories \(\text{Br}_B(G)\) and \(\text{Br}_b(C_G(S))\) are equivalent. Moreover the blocks \(B\) and \(b\) are isotypic. -- The correspondences in Theorem 2 are obtained from Theorem 1. The final Section 4 of this paper uses Shintani descent to obtain a perfect isometry. Let \(\widehat L\) be a finite field of characteristic \(\ell\), let \(\sigma\) be a Frobenius automorphism of \(\widehat L\) and let \(C_{\widehat L}(\sigma)=L\) where \([\widehat L:L]=m\). Here \(\sigma\) acts naturally on \(\text{GL}_n(\widehat L)\) with fixed points \(\text{GL}_n(L)\). Let \(B(\text{GL}_n(\widehat L))\) and \(B(\text{GL}_n(L))\) denote the principal blocks of \(\text{GL}_n(\widehat L)\) and \(\text{GL}_n(L)\) resp. Shintani descent is…
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1340590
zbMATH Keywords: \(p\)-modular systems
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: finite groups
zbMATH Keywords: central idempotents
zbMATH Keywords: irreducible characters
zbMATH Keywords: Glauberman correspondence
zbMATH Keywords: defect groups
zbMATH Keywords: perfect isometries
zbMATH Keywords: Brauer categories
zbMATH Keywords: Shintani descent
zbMATH Keywords: principal blocks
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1998.7779
OpenAlex ID: W1997330893
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DOI: 10.1006/JABR.1998.7779
description: scientific article; zbMATH DE number 1340590