Quasi-hereditary property of double Burnside algebras

Abstract

In this short note, we investigate some consequences of the vanishing of simple biset functors. As a corollary, if there is no non-trivial vanishing of simple biset functors (e.g., if the group G is commutative), then we show that kB(G, G) is a quasi-hereditary algebra in characteristic zero. In general, this is 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.

Around evaluations of biset functors

“…Indeed, as explained in Section 9 of [7], the evaluation at a group G of the radical of a biset functor is not always the radical of the evaluation. In [21], we observed that this phenomenon is connected with the vanishing property of the simple biset functors. Lemma 3.2.…”

“…In particular, it may be possible to find a more suitable order on the set of simple modules over the double Burnside algebra in order to avoid the vanishing problems. In [21], we proved that the global dimension of CB(A 5 , A 5 ) is infinite. In particular, this shows that such a better ordering does not exist for the double Burnside algebra for A 5 .…”