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arXiv · 1502.04633

Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

Samuel Clearman·Matthew Hyatt·Brittany Shelton·Mark Skandera
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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.

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Samuel Clearman, Matthew Hyatt, Brittany Shelton, Mark Skandera. 2015-02-13. Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements. https://arxiv.org/abs/1502.04633

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