The module of affine descents

Abstract

The goal of this paper is to introduce an algebraic structure on the space spanned by affine descent classes of a Weyl group, by analogy and in relation to the structure carried by ordinary descent classes. The latter classes span a subalgebra of the group algebra, Solomon's descent algebra. We show that the former span a left module over this algebra. The structure is obtained from geometric considerations involving hyperplane arrangements. We provide a combinatorial model for the case of the symmetric group.

Keywords

• weyl group
• coxeter complex
• hyperplane arrangement
• tits product
• solomon's descent algebra
• steinberg torus
• [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]