On weak crossed products, Frobenius algebras, and the weak Bruhat ordering

The weak crossed product algebra was studied first by Haile, Larson, and Sweedler [Amer. J. Math. 105 (1983) 689]. They gave interesting properties for such an algebra, one of which is that if Af is a weak crossed product induced by a weak 2-cocycle f defined on a Galois group G=Gal(K/F), and H is the inertial subgroup of G, then Af has a Wedderburn splitting, that is Af=B J where J is the radical of Af and B is a KH-central simple algebra. The purpose of this paper is to give the necessary and sufficient condition for a weak crossed product to be Frobenius and to describe an algorithm for constructing lower subtractive graphs from a finite group G and a generating set S. A special case of this construction is the so-called weak Bruhat ordering on a Coxeter group (G,S). We show that the nilCoxeter algebra associated to (G,S) is a special case of the restricted algebra associated to a lower subtractive graph.