Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

Samuel Clearman¹ ; Matthew Hyatt² ; Brittany Shelton³ ; Mark Skandera¹

¹Lehigh University
²Pace University
³Albright College

The electronic journal of combinatorics, Tome 23 (2016) no. 2

Résumé

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 $\smash{\chi_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, $\smash{\epsilon_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, and $\smash{\eta_q^\lambda(q^{\ell(w)/2} C'_w(q))}$. 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.

Détail

DOI : 10.37236/5021

Classification : 05E10, 20C08, 05E05
Mots-clés : Hecke algebra trace, Kazhdan-Lusztig basis, chromatic quasisymmetric function, \(P\)-tableau, planar network, pattern avoidance