Categorifying induction formulae via divergent series

TL;DR

This paper develops explicit induction formulae for finite group representations and rational Green functors by summing divergent series over specialized decomposition spaces, resulting in new and known formulas with rational coefficients.

Contribution

It introduces a novel method of summing divergent series over Dwyer's spaces to derive explicit induction formulae, including new formulas in the context of Green functors.

Findings

Derived explicit induction formulae with rational coefficients.

Established new induction formulae from Dwyer's subgroup and centralizer spaces.

Connected the results to group cohomology and classifying space splittings.

Abstract

We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.