Equivariant \(K\)-theory and Alperin's conjecture
type: Item
instance of: scholarly article
title: Equivariant \(K\)-theory and Alperin's conjecture
zbMATH Open document ID: 0781.20012
DOI: 10.1016/0022-4049(93)90052-U
author: Jacques Thévenaz
published in: Journal of Pure and Applied Algebra
publication date: 1993-06-29T00:00:00Z
review text: Alperin's weight conjecture in modular representation theory has inspired a lot of work in terms of reformulations and generalizations. This paper presents a conjecture about the equivariant \(K\)-theory of the simplicial complex of \(p\)-subgroups of a finite group \(G\), which is equivalent to Alperin's conjecture. This provides a new point of view to Alperin's conjecture, but does not seem to bring a solution nearer. The conjecture is described as follows: Let \(G\) be a finite group and \(p\) a prime number. Brown's complex (for \(G\) and \(p\)) is the simplicial complex \({\mathcal S}_ p(G)\) whose set of \(k\)-simplices consists of all chains \(\sigma = (P_ 0 < P_ 1 < \dots < P_ k)\) of non-trivial \(p\)- subgroups of \(G\) (the faces of \(\sigma\) being the shorter chains). This is an ordered simplicial complex which is finite, so that its geometric realization \(| {\mathcal S}_ p(G)|\) is a compact subset of some Euclidean space. We view \(|{\mathcal S}_ p(G)|\) as a compact \(G\)- space. Conjecture: \(\chi_ G({\mathcal S}_ p(G))=k(G) - z_ p(G)\). (Here \(\chi_ G\) denotes the `equivariant Euler characteristic', \(k(G)\) the number of irreducible complex representations of \(G\) and \(z_ p(G)\) the number of \(p\)-blocks of defect 0 in \(G\)).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 19D10
zbMATH DE Number: 221972
zbMATH Keywords: Alperin's weight conjecture
zbMATH Keywords: modular representation
zbMATH Keywords: equivariant \(K\)- theory
zbMATH Keywords: simplicial complex of \(p\)-subgroups
zbMATH Keywords: finite group
zbMATH Keywords: ordered simplicial complex
zbMATH Keywords: irreducible complex representations
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: defect
Wikidata QID: Q122860112
MaRDI profile type: Publication
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full work available at URL: 0022-4049(93)90052-u
OpenAlex ID: W1969135049
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description: scientific article; zbMATH DE number 221972