Fusion category algebras.
type: Item
instance of: scholarly article
title: Fusion category algebras.
zbMATH Open document ID: 1055.20009
author: Markus Linckelmann
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: Alperin's weight conjecture is one of the main open problems in representation theory of finite groups. Many reformulations of this conjecture exist. The paper under review presents another such reformulation. This reformulation is an equality between the number of simple modules of a \(p\)-block \(b\) of a finite group \(G\) and the number of simple modules of a certain quasi-hereditary algebra \(k\overline{\mathcal F}(b)\) over a suitable field \(k\) of characteristic \(p>0\). The construction of \(k\overline{\mathcal F}(b)\) involves the fusion system \(\mathcal F\) of \(b\), a category whose objects are the subgroups of a defect group \(P\) of \(b\). The morphisms between two objects \(Q\) and \(R\) of \(\mathcal F\) are the restrictions of certain inner automorphisms of \(G\). The construction of \(k\overline{\mathcal F}(b)\) depends also on a certain 2-cocycle \(\alpha\) on \(\mathcal F\). The author leaves the existence of \(\alpha\) open. However, Puig has pointed out that the existence of \(\alpha\) follows from unpublished work of his.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16S99
zbMATH DE Number: 2102476
zbMATH Keywords: blocks
zbMATH Keywords: defect groups
zbMATH Keywords: Brauer pairs
zbMATH Keywords: fusion systems
zbMATH Keywords: Alperin weight conjecture
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: numbers of simple modules
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.12.010
OpenAlex ID: W2035441577
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DOI: 10.1016/J.JALGEBRA.2003.12.010
description: scientific article; zbMATH DE number 2102476