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

2022/04/04 by Kim, Jang Soo, Yun, Sun-mi
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.01310

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.

Related

• The characteristic polynomials of imprimitive groups and affine Coxeter groups
• Characteristic polynomials and some combinatorics for finite Coxeter groups
• A new family of posets generalizing the weak order on some Coxeter groups
• Characteristic Polynomials of Deformations of Coxeter Arrangements via levels of regions
• The absolute order of a permutation representation of a Coxeter group