Around evaluations of biset functors

TL;DR

This paper investigates the structure of the double Burnside algebra for non-vanishing groups, establishing an equivalence with biset functors and analyzing its algebraic properties over fields.

Contribution

It demonstrates an equivalence between module categories and biset functors, and characterizes the algebra's quasi-hereditary and self-injective properties.

Findings

Double Burnside algebra is quasi-hereditary over characteristic zero fields.

The algebra is self-injective if and only if it is semisimple.

Evaluation functor induces an equivalence for non-vanishing groups.

Abstract

For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.