Quasi-hereditary structure of twisted split category algebras revisited

Abstract

Available online xxxx Communicated by Michel Broué MSC: 16G10 20M17 19A22 Keywords: Split category Regular monoid Quasi-hereditary algebra Highest weight category Biset functor Brauer algebraLet k be a field of characteristic 0, let C be a finite split category, let α be a 2-cocycle of C with values in the multiplicative group of k, and consider the resulting twisted category algebra A := k α C. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over k is equivalent to a module category over a condensed algebra εAε, for an idempotent ε of A. In [2] the authors proved that A is quasi-hereditary (with respect to an explicit partial order on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order by introducing a coarser order P leading to the same results on A, but which allows to pass the quasi-heredity result to the condensed algebra εAε describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [21]. The new partial order P has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.