A new approach to Glauberman's correspondence.
type: Item
instance of: scholarly article
title: A new approach to Glauberman's correspondence.
zbMATH Open document ID: 1037.20004
author: Everett C. Dade
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: This interesting paper generalizes the Glauberman character correspondence of finite group characters to twisted group algebras. Special cases of this result include the Glauberman correspondence and correspondences similar to those obtained by other authors. Let \(E\) be a finite group with a normal subgroup \(G\) such that \(F=E/G\) is cyclic. Let \(\mathcal A\) be a twisted group algebra of \(E\) over a field \(\mathcal F\) of characteristic zero that splits \({\mathcal A}[H]\) for each \(H\leq E\). For any such \(H\leq E\), the set \(\text{Ch}({\mathcal A}[H])\) of all virtual \(\mathcal F\)-characters of \({\mathcal A}[H]\) is a free \(\mathbb{Z}\)-module with basis \(\text{Irr}({\mathcal A}[H])\). The dual \(\widehat F\) of all linear \(\mathcal F\)-characters of \(F=E/G\) permutes \(\text{Irr}({\mathcal A}[H])\) and so \(\text{Ch}({\mathcal A}[H])\) becomes a \(\mathbb{Z} F\)-module and \(\widehat F\) preserves the natural inner product \((\phi,\psi)_{{\mathcal A}[H]}\in\mathbb{Z}\) of characters \(\phi,\psi\in\text{Ch}({\mathcal A}[H])\). A \(\mathbb{Z}\widehat F\)-submodule \(\text{Ch}({\mathcal A}[E]| E_0)\) of \(\text{Ch}({\mathcal A}[E])\) is defined, let \(\mu=\min\{(\theta,\theta)_{{\mathcal A}[E]}\mid\theta\in\text{Ch}({\mathcal A}[E]| E_0)\}\) and set \(\min\text{Ch}({\mathcal A}[E]| E_0)=\{\theta\in\text{Ch}({\mathcal A}[E]| E_0)\mid(\theta,\theta)=\mu\}\). Clearly \(\min\text{Ch}({\mathcal A}[E]| E_0)\) is \(\widehat F\)-stable. Two characters \(\phi,\psi\in\min\text{Ch}({\mathcal A}[E]| E_0)\) are ``connected'' if \((\phi,\lambda\psi)_{{\mathcal A}[E]}\neq 0\) for some \(\lambda\in\widehat F\). The fundamental result (Theorem 4.7) of this paper proves that this connectivity is an equivalence relation and that its equivalence classes biject with \(\text{Irr}^E({\mathcal A}[G])\), the \(E\)-stable characters of \(\text{Irr}({\mathcal A}[G])\). Hence any isomorphism preserving the relevant structures induces a bijection of characters. Sections 6-10 o…
reviewed by: Morton E. Harris
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 2037078
zbMATH Keywords: Glauberman character correspondence
zbMATH Keywords: twisted group algebras
zbMATH Keywords: virtual characters
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2002.11.002
OpenAlex ID: W2080464432
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DOI: 10.1016/J.JALGEBRA.2002.11.002
description: scientific article; zbMATH DE number 2037078