Relations between permutation representations in positive characteristic

TL;DR

This paper investigates the structure of the kernel of the permutation module map from the Burnside ring to the representation ring over a field of positive characteristic, extending known results from characteristic zero to most cases in characteristic p>0.

Contribution

It provides a description of the kernel K_F(G) for fields of characteristic p>0, except in the complex case involving non-p-hypo-elementary (p,p)-Dress groups.

Findings

Complete description of K_F(G) in positive characteristic for most groups.

Extension of known characteristic zero results to positive characteristic.

Identification of the complex case involving non-p-hypo-elementary (p,p)-Dress groups.

Abstract

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic p>0 in all but the most complicated case, which is when G has a subquotient that is a non-p-hypo-elementary (p,p)-Dress group.