Tornehave morphisms. III: The reduced tornehave morphism and the Burnside unit functor.
Let \(B(G)\) be the Burnside ring of a finite group \(G\) and, for \(\mathbb{K} \in \{\mathbb{Q},\mathbb{R}\}\), let \(A_{\mathbb{K}}(G)\) be the ring of representations of \(G\) over \(\mathbb{K}\). Moreover, let \(K(G)\) be the kernel of the linearization map \(B(G) \to A_{\mathbb{Q}}(G)\), and let \(K^{\times}(G)\) be the kernel of its restriction \(B^{\times}(G) \to A^{\times}_{\mathbb{Q}}(G)\) where \(B^{\times}(G)\) and \(A^{\times}_{\mathbb{Q}}(G)\) denote the groups of units of \(B(G)\) and \(A_{\mathbb{Q}}(G)\). Furthermore, the image of the tom Dieck map \(\overline{\text{die}}_G:A_{\mathbb{R}}(G) \to B^{\times}(G)\) is denoted by \(^{\mathbb{R}}B^{\times}(G)\), and the image of \(A_{\mathbb{Q}}(G)\), viewed as a subring of \(A_{\mathbb{R}}(G)\), under \(\overline{\text{die}}_G\) is denoted by \(^{\mathbb{Q}}B^{\times}(G)\). The author proves that \(B^{\times}(G)=~ ^{\mathbb{R}}B^{\times}(G)\) if \(G\) is nilpotent, and that \(B^{\times}(G)=K^{\times}(G)+~ ^{\mathbb{Q}}B^{\times}(G)\) in general. These facts are due to Tornehave. For a set \(\pi\) of prime numbers, there is a reduced Tornehave map \(\overline{\text{torn}}^{\pi}_G :K(G) \to B^{\times}(G)\). The author shows that, for a prime \(p\) and a finite \(p\)-group \(P\), the map \(\overline{\text{torn}}^{\{p\}}_P\) induces an isomorphism \(K(P)/^{\mathbb{Q}}K(P) \to B^{\times}(P)/^{\mathbb{Q}}B^{\times}(P)\) first defined by Bouc; here \(^{\mathbb{Q}}K(P)\) denotes a specific subgroup of \(K(P)\) whose definition we omit. For Parts I, II, see [the author, Commun. Algebra 39, No. 1, 355--395 (2011; Zbl 1211.19001); J. Pure Appl. Algebra 214, No. 10, 1759--1777 (2010; Zbl 1225.19001)].
scientific article; zbMATH DE number 6509808
scientific article