Permutation modules, \(p\)-permutation modules and Conlon species
type: Item
instance of: scholarly article
title: Permutation modules, \(p\)-permutation modules and Conlon species
zbMATH Open document ID: 0785.20008
author: Yun Fan
published in: Science in China. Series A
publication date: 1993-01-17T00:00:00Z
review text: Let \(F\) be a field of prime characteristic \(p\), let \(G\) be a finite group, and let \(a(FG,\text{Triv})\) denote the \(\mathbb{Z}\)-span of all isomorphism classes of \(p\)-permutation \(FG\)-modules. Similarly, \(a(FG,\text{Perm})\) denotes the \(\mathbb{Z}\)-span of all isomorphism classes of permutation \(FG\)-modules. Thus \(a(FG,\text{Triv})\) and \(a(FG,\text{Perm})\) are subrings of the Green ring \(a(FG)\). We set \(A(FG,\text{Triv}):= \mathbb{Q}\otimes a(FG,\text{Triv})\) and \(A(FG,\text{Perm}):= \mathbb{Q}\otimes a(FG,\text{Perm})\). The species \(A(FG,\text{Triv})\to\mathbb{C}\) are determined explicitly. They take values in a suitable cyclotomic extension \(\mathbb{Q}(\varepsilon)\) of \(\mathbb{Q}\). The Galois group \(\Gamma\) of \(\mathbb{Q}(\varepsilon)|\mathbb{Q}\) thus acts (pointwise) on these species. This induces an action of \(\Gamma\) on \(\mathbb{Q}(\varepsilon)\otimes A(FG,\text{Triv})\) leaving invariant \(a(FG,\text{Triv})\). The author shows that for the fixed points of this action one has \(| G|\cdot a(FG,\text{Triv})^ \Gamma\subseteq a(FG,\text{Perm}) \subseteq a(FG,\text{Triv})\). Thus, in particular, \(A(FG,\text{Perm})= A(FG,\text{Triv})^ \Gamma\).
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 98014
zbMATH Keywords: finite group
zbMATH Keywords: isomorphism classes of \(p\)-permutation \(FG\)-modules
zbMATH Keywords: permutation \(FG\)-modules
zbMATH Keywords: Green ring
zbMATH Keywords: species
zbMATH Keywords: action
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 98014