Permutation modules, \(p\)-permutation modules and Conlon species
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, \(A(FG,\text{Perm})= A(FG,\text{Triv})^ \Gamma\).
isomorphism classes of \(p\)-permutation \(FG\)-modules
scientific article; zbMATH DE number 98014