Equivariant \(K\)-theory and Alperin's conjecture
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 \(G\)).
scientific article; zbMATH DE number 221972
scientific article