Simple biset functors and double Burnside rings

TL;DR

This paper investigates simple modules of the double Burnside ring for finite groups, showing they are evaluations of simple biset functors, and introduces a bilinear form to analyze their structure.

Contribution

It introduces a bilinear form on biset modules and characterizes simple modules of the double Burnside ring as evaluations of simple biset functors.

Findings

Simple modules correspond to evaluations of simple biset functors.

A bilinear form helps identify the semi-simple quotient of biset functors.

The structure of simple modules is explicitly described.

Abstract

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(-,H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.