Descent representations and multivariate statistics

Abstract

Combinatorial identities on Weyl groups of types A A and B B are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type A A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B B , using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.