Generalised Burnside Rings, G-categories and Module Categories

TL;DR

This paper applies generalized Burnside rings to algebraic representation theory, providing explicit tables for S_4 and S_5, and computing base sets for a specific nilpotent element in F_4.

Contribution

It introduces an application of generalized Burnside rings to algebraic representation theory with explicit computations for certain symmetric groups.

Findings

Explicit tables of marks for S_4 and S_5

Computed base sets for F_4 (a_3) nilpotent element

Demonstrated relevance to reductive algebraic groups

Abstract

This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups $S_4$ and $S_5$ which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element $F_4 (a_3)$ are computed.