Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

TL;DR

This paper provides new combinatorial and algebraic interpretations of Hecke algebra traces at Kazhdan-Lusztig basis elements, linking them to chromatic quasisymmetric functions and confirming a conjecture by Haiman.

Contribution

It introduces novel combinatorial interpretations of Hecke algebra traces at specific basis elements and connects these to chromatic quasisymmetric functions, expanding understanding of Hecke algebra representations.

Findings

Combinatorial interpretation of certain Hecke algebra trace polynomials.

New algebraic interpretation of chromatic quasisymmetric functions.

Confirmation of Haiman's conjectured formula.

Abstract

For irreducible characters $\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$, induced sign characters $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, and induced trivial characters $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $\chi_q^\lambda(q^{l(w)/2}C'_w(q))$, $\epsilon_q^\lambda(q^{l(w)/2} C'_w(q))$, and $\smash{\eta_q^\lambda(q^{l(w)/2} C'_w(q))}$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of results of Haiman. We prove similar results for other $H_n(q)$-traces, and confirm a formula conjectured by Haiman.