Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal \(p\)-Sylow subgroup
type: Item
instance of: scholarly article
title: Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal \(p\)-Sylow subgroup
zbMATH Open document ID: 0814.20006
DOI: 10.1016/S0021-8693(05)80033-X
author: Alfred Wiedemann
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic \(p > 0\). Let \(G\) be a finite group with a cyclic normal \(p\)-Sylow subgroup \(P\), \(H\) be a complement in \(G\). The Mackey algebra is the basic \(k\)-algebra whose category of finitely generated modules is equivalent to the category of Mackey functors for \(G\) with the property that all \(k\)-spaces involved are finite dimensional over \(k\). The simple Mackey functors for \(G\) are in one-to-one correspondence to the conjugacy classes of pairs \((G',E)\), \(G' \leq G\), \(E\) a simple \(k(N_ G(G')/G')\)-module. The author proves that the simple Mackey functors associated with those subgroups of \(G\) which are not contained in \(Z = P \times C_ H P\) are projective and injective. The simple Mackey functors associated with the subgroups of \(Z\) form in the quiver of the Mackey algebra connected components of a special form. This form is explicitly computed by the author.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 130635
zbMATH Keywords: complements
zbMATH Keywords: finite group
zbMATH Keywords: cyclic normal \(p\)-Sylow subgroup
zbMATH Keywords: Mackey algebra
zbMATH Keywords: basic \(k\)-algebra
zbMATH Keywords: category of finitely generated modules
zbMATH Keywords: category of Mackey functors
zbMATH Keywords: simple Mackey functors
zbMATH Keywords: quiver
reviewed by: S. D. Kozlov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130635