Representation groups for the Schur index
type: Item
instance of: scholarly article
title: Representation groups for the Schur index
zbMATH Open document ID: 0571.20005
DOI: 10.1016/0021-8693(85)90076-6
author: Peter Schmid
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G be a finite group, \(\chi\) an irreducible complex character of G and K an algebraic number field containing the values of \(\chi\). Then \(\chi\) determines a central simple component A in the group algebra KG and thus an element [A] in the Brauer group Br(K). By class field theory the order of A in Br(K) coincides with the Schur index \(m_ K(\chi)\). As usual, Br(K) can be identified with \(H^ 2(\Gamma,\bar K^{\times})\) where \(\bar K\) denotes the algebraic closure of K and \(\Gamma\) the Galois group of \(\bar K\) over K. The author constructs, for any prime p, a group extension \(1\to M\to P\to P/M\to 1\) inside G, an epimorphism \(\Gamma\) \(\to P/M\) and a compatible homomorphism \(M\to \bar K^{\times}\) such that the corresponding change of groups \(H^ 2(P/M,M)\to H^ 2(\Gamma,\bar K^{\times})\) maps the cohomology class of \(1\to M\to P\to P/M\to 1\) onto the p-component \([A]_ p\) of [A]. This method gives a unifying approach to several results on the Schur index.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 3910594
zbMATH Keywords: finite group
zbMATH Keywords: irreducible complex character
zbMATH Keywords: central simple component
zbMATH Keywords: group algebra
zbMATH Keywords: Brauer group
zbMATH Keywords: Schur index
zbMATH Keywords: group extension
zbMATH Keywords: cohomology
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90076-6
OpenAlex ID: W2012195912
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description: scientific article; zbMATH DE number 3910594