Evaluations of Hecke Algebra Traces at Kazhdan-Lusztig Basis Elements

  • Samuel Clearman
  • Matthew Hyatt
  • Brittany Shelton
  • Mark Skandera
Keywords: Hecke algebra trace, Kazhdan-Lusztig basis, Chromatic quasisymmetric function, P-tableau, Planar network, Pattern avoidance


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.

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