Characteristic polynomials of the weak order on classical and affine Coxeter groups

TL;DR

This paper derives simple product formulas for the characteristic polynomials of permutations with fixed descent sets, extends these results to Coxeter groups, and explores generating functions relating classical and affine Coxeter groups.

Contribution

It introduces new product formulas for characteristic polynomials in the context of Coxeter groups and their weak order, generalizing previous results for permutations.

Findings

Product formula for fixed descent set permutations

Characteristic polynomial for alternating permutations

Generating function linking classical and affine Coxeter groups

Abstract

We find a simple product formula for the characteristic polynomial of the permutations with a fixed descent set under the weak order. As a corollary we obtain a simple product formula for the characteristic polynomial of alternating permutations. We generalize these results to Coxeter groups. We also find a formula for the generating function for the characteristic polynomials of classical Coxeter groups, which is then related to affine Coxeter groups.