A note on Green functors with inflation

TL;DR

This paper axiomatizes Green functors with inflation to understand when G-sets induce isomorphic representations, unifying and extending several known results in representation theory and Brauer relations.

Contribution

It introduces a general axiomatic framework for Green functors with inflation, encompassing and extending previous results on monomial and Brauer relations.

Findings

Unified framework for Green functors with inflation

Generalized results on Brauer relations in characteristic p>0

Includes special cases like Deligne's monomial relations

Abstract

This note is motivated by the problem to understand, given a commutative ring F, which G-sets X, Y give rise to isomorphic F[G]-representations F[X]\cong F[Y]. A typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of G. In the present paper we axiomatise the situation, and prove such a result in the generality of Mackey functors and Green functors with inflation. Our result includes, as special cases, a result of Deligne on monomial relations, a result of the first author and Tim Dokchitser on Brauer relations in characteristic 0, and a new result on Brauer relations in characteristic p>0. We will need the new result in a forthcoming paper on Brauer relations in positive characteristic.