Brauer-Clifford equivalence of full matrix algebras.
type: Item
instance of: scholarly article
title: Brauer-Clifford equivalence of full matrix algebras.
zbMATH Open document ID: 1186.20012
author: Alexandre Turull
published in: Journal of Algebra
publication date: 2009-09-10T00:00:00Z
review text: Let \(G\) be a finite group, \(F\) a field, and \(Z\) a commutative central simple \(G\)-algebra. In [J. Algebra 321, No. 12, 3620-3642 (2009; Zbl 1186.20011)], the author introduced the Brauer-Clifford group \(\text{BrCliff}(G,Z)\). This group consists of certain equivalence classes of central simple \(G\)-algebras, and the product is associated with the tensor product of algebras. In [op. cit.] it was proved that it has a natural homomorphism into a Brauer group, and that the kernel of this homomorphism is the set of all equivalence classes of \(G\)-algebras which are full matrix algebras. In the paper under review it is shown that this kernel is isomorphic to a certain second cohomology group. In certain situations it is also shown how to compute the element of the second cohomology group associated with a certain family of characters. The results in this paper are used in the author's proof of a strengthening of the McKay conjecture for \(p\)-solvable groups [in Adv. Math. 217, No. 5, 2170-2205 (2008; Zbl 1149.20010)].
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 5602623
zbMATH Keywords: Clifford theory
zbMATH Keywords: Brauer groups
zbMATH Keywords: Schur indices
zbMATH Keywords: finite groups
zbMATH Keywords: representations
zbMATH Keywords: central simple algebras
zbMATH Keywords: cohomology groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.02.018
OpenAlex ID: W2061808938
cites work: Cohomology of modules in the principal block of a finite group
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cites work: Q3961610
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cites work: The Brauer-Clifford group.
DOI: 10.1016/J.JALGEBRA.2009.02.018
description: scientific article; zbMATH DE number 5602623