Shintani descent and perfect isometries for blocks of finite general linear groups.
type: Item
instance of: scholarly article
title: Shintani descent and perfect isometries for blocks of finite general linear groups.
zbMATH Open document ID: 1078.20013
author: Radha Kessar
published in: Journal of Algebra
publication date: 2004-11-19T00:00:00Z
review text: Let \(q\) be a prime power and let \(n,d\) be positive integers. Let \(\mathcal F\) denote the algebraic closure of the field of \(q\) elements and let \(V\) be an \(n\)-dimensional vector space over \(\mathcal F\). Set \({\mathcal G}=\text{GL}_n({\mathcal F})=\text{GL}(V,{\mathcal F})\) (the general linear group of \(V\) over \(\mathcal F\)). Let \(F\colon{\mathcal G}\to{\mathcal G}\) denote the standard Frobenius morphism on \(\mathcal G\) corresponding to an \({\mathcal F}_q\)-rational structure for \(\mathcal G\) such that \({\mathcal G}^F\cong\text{GL}_n(q)\). Clearly \({\mathcal G}^{F^d}\cong\text{GL}_n(q^d)\). Let \(\sigma\colon{\mathcal G}^{F^d}\to{\mathcal G}^{F^d}\) be the automorphism of \({\mathcal G}^{F^d}\) obtained by restriction of \(F\). Set \(\Gamma={\mathcal G}^{F^d}\rtimes\langle\sigma\rangle\) and let \(K\) denote a field of characteristic zero that is a splitting field for \({\mathcal G}^F\) and \({\mathcal G}^{F^d}\). Let \({\mathcal C}({\mathcal G}^F)\) denote the set of \(K\)-valued class functions on \({\mathcal G}^F\). Note that the subset \({\mathcal G}^{F^d}\sigma\) of \(\Gamma\) is invariant under conjugation by elements of \(\Gamma\). Two elements \(g,h\) of \({\mathcal G}^{F^d}\) are called \(\sigma\)-conjugate if \(h=xg^\sigma x^{-1}\) for some \(x\in{\mathcal G}^{F^d}\). There is a well-defined bijective map, \(N_{F^d/F}\), that sends \(\sigma\)-conjugacy classes of \({\mathcal G}^{F^d}\) into conjugacy classes of \({\mathcal G}^F\). Also, if \(g,h\in{\mathcal G}^{F^d}\), then \(g\sigma\) and \(h\sigma\) are conjugate in \(\Gamma\) if and only if \(g\) and \(h\) are \(\sigma\)-conjugate in \({\mathcal G}^{F^d}\). The set of functions on \({\mathcal G}^{F^d}\sigma\) obtained by restriction of \(K\)-valued class functions on \(\Gamma\) is denoted by \({\mathcal C}({\mathcal G}^{F^d}\sigma)\). The map \(\text{Sh}_{F^d/F}\colon{\mathcal C}({\mathcal G}^{F^d}\sigma)\to{\mathcal C}({\mathcal G}^F)\) defined by: \(\text{Sh}_{F^d/F}(f)(N_{F…
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 2116870
zbMATH Keywords: general linear groups
zbMATH Keywords: class functions
zbMATH Keywords: irreducible characters
zbMATH Keywords: modular systems
zbMATH Keywords: unipotent blocks
zbMATH Keywords: perfect isometries
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.013
OpenAlex ID: W2022992703
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DOI: 10.1016/J.JALGEBRA.2004.02.013
description: scientific article; zbMATH DE number 2116870