The Steinberg torus of a Weyl group as a module over the Coxeter complex

TL;DR

This paper constructs a module structure on the Steinberg torus associated with a root system over the Coxeter complex, linking geometric hyperplane arrangements with algebraic structures in Weyl groups.

Contribution

It introduces a novel module structure on the faces of the Steinberg torus over the Coxeter complex, connecting geometric and algebraic aspects of Weyl groups.

Findings

Module structure on faces of Steinberg torus established

Connection between affine and ordinary descent classes demonstrated

Combinatorial models provided for types A and C

Abstract

Associated to each irreducible crystallographic root system $\Phi$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $\Phi$. This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of $\Phi$. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space spanned by affine descent classes of a Weyl group, over the space spanned by ordinary descent classes. The latter constitute a subalgebra of the group algebra, the classical descent algebra of Solomon. We provide combinatorial models for the module of faces when $\Phi$ is of type $A$ or $C$.