Quasi-hereditary property of double Burnside algebras

TL;DR

This paper explores conditions under which double Burnside algebras are quasi-hereditary, showing that in characteristic zero, they are if simple biset functors do not vanish, with counterexamples when they do.

Contribution

It establishes a link between the vanishing of simple biset functors and the quasi-hereditary property of double Burnside algebras in characteristic zero.

Findings

Double Burnside algebra is quasi-hereditary if no non-trivial simple biset functor vanishes.

Counterexample: the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.

In characteristic zero, the non-vanishing condition ensures quasi-hereditary structure.

Abstract

In this short note we investigate some consequences of the vanishing of simple biset functors. As corollary, if there is no non-trivial vanishing of simple biset functors (e.g. if the group is commutative), then we show that the double Burnside algebra is a quasi-hereditary algebra in characteristic zero. In general, this not true without the non-vanishing condition, as over a field of characteristic zero, the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.