Around evaluations of biset functors (Q2421927)

Let \(k\) be a field and \(G\) be a finite group. The double Burnside algebra of \(G\) is the algebra algebra \(kB(G,G)=k\otimes_{\mathbb{Z}}B(G,G)\), where \(B(G,G)\) is the Grothendieck ring of the category of all finite \(G\)-\(G\)-bisets. The aim of this paper is to study double Burnside algebras via evaluations of biset functors. However, there is problem with evaluating simple biset functors: The evaluation at \(G\) of a simple biset functor \(S\) is either \(0\) or a simple module over the endomorphism algebra of \(G\). The simple functors are indexed by a minimal group \(H\) and a simple \(k \mathrm{Out}(H)\)-module. If \(H\) is not isomorphic to a subquotient of G, then the corresponding simple functor vanishes at \(G\), and this is called a trivial vanishing. If \(H\) is a subquotient of \(G\), it is still possible that the simple functor vanishes at \(G\), and this is then called a non-trivial vanishing. Therefore, the author looks at non-vanishing groups, that is, finite groups for which there is no non-trivial vanishing. He shows that the abelian groups and the self-dual groups are non-vanishing, and there are infinitely many others, but a classification of all the non-vanishing groups is not achieved. He proves that for a non-vanishing group, there is an equivalence between the category of modules over \(kB(G,G)\) and a specific category of biset functors. The author shows that in characteristic zero, \(kB(G,G)\) is quasihereditary when \(G\) is a non-vanishing group, and results about the highest weight structure are deduced. It is known that for a quasi-hereditary \(A\), there is a Morita equivalent algebra \(A'\) which has an exact Borel subalgebra. It turns out here that Borel subalgebras are connected with the deflation functors. It is shown in this paper that the category of deflation functors is an exact Borel subcategory of the category of biset functors. In the final section, Barker's theorem about the semi-simple property of the category of biset functors is revisited. Using the characterization of the semi-simplicity by the trivial object, the author shows that the \(kB(G,G)\) is not a self-injective algebra, except when it is semi-simple.