Brauer relations for finite groups in the ring of semisimplified modular representations

TL;DR

This paper investigates the kernel of a map from the Burnside ring to the Grothendieck ring of modular representations, classifying primitive quotients for all finite groups and identifying cases with non-trivial quotients.

Contribution

It provides a complete classification of the primitive quotient of the kernel for all finite groups and describes generators in the soluble case, advancing understanding of modular representation relations.

Findings

Classified primitive quotients for all finite groups.

Identified groups with non-trivial primitive quotient.

Provided generators for the primitive quotient in soluble groups.

Abstract

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for all finite groups, to classify the primitive quotient of the kernel; that is for each $G$, the kernel modulo elements coming from the kernel for proper subquotients of $G$. We are able to identify exactly which groups have non-trivial primitive quotient and we give generators for the primitive quotient in the soluble case.