Quasi-hereditary property of double Burnside algebras

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.

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