Algebraicisation of explicit Brauer induction (Q1188169)

Brauer induction for the representations of a finite group \(G\) may be described as follows. Let \(R(G)\) be the complex representation ring of \(G\), and let \(R_ +(G)\) be the free abelian group generated by \(G\)- conjugacy classes of linear complex representations \(\phi\) of subgroups of \(G\). If \(\phi\) is defined on \(H\), put \(b_ G(\phi) = \text{Ind}^ G_ H(\phi)\). Then \(b_ G: R_ +(G) \to R(G)\) is a surjection. By ``an explicit Brauer induction formula'', the authors means the specification of a section for \(b_ G\). In a previous paper, \textit{V. Snaith} [Invent. Math. 94, 455-478 (1988; Zbl 0704.20009)] gives a topological construction for such a section, \(t_ G\). The aim of this paper is to give an algebraic derivation of \(t_ G\); the methods are too technical to discuss here. The authors show that, although \(t_ G\) is not a homomorphism, it is closely related to a homomorphic section \(a_ G\) defined by \textit{R. Boltje} [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)].