Quasi-hereditary structure of twisted split category algebras revisited

摘要

Let k be a field of characteristic 0, let C be a finite split category, let cc be a 2-cocycle of C with values in the multiplicative group of k, and consider the resulting twisted category algebra A := k(alpha)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 epsilon A epsilon, for an idempotent epsilon 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 <= 1 leading to the same results on A, but which allows to pass the quasi-heredity result to the condensed algebra epsilon A epsilon describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [21]. The new partial order has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset fimctors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules. (C) 2015 Elsevier Inc. All rights reserved.