Affine descents and the Steinberg torus

TL;DR

This paper investigates the combinatorial properties of the Steinberg torus derived from affine Weyl groups, revealing symmetry, unimodality, and generating functions related to descent statistics.

Contribution

It establishes the connection between the $h$-polynomials of the Steinberg torus and descent-like statistics, including nonnegativity and symmetry properties.

Findings

The $h$-polynomials are generating functions over $W$ for a descent statistic.

The $h$-polynomial has a nonnegative $ ext{gamma}$-vector, indicating symmetry and unimodality.

Explicit expansions and identities are provided for classical cases.

Abstract

Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.