# Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

Abstract : For irreducible characters $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ and induced sign characters $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the pattern 312, we combinatorially interpret the polynomials $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ and $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$. This gives a new algebraic interpretation of $q$-chromatic symmetric functions of Shareshian and Wachs. We conjecture similar interpretations and generating functions corresponding to other $H_n(q)$-traces.
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Conference papers

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### Citation

Sam Clearman, Matthew Hyatt, Brittany Shelton, Mark Skandera. Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.1089-1100, ⟨10.46298/dmtcs.2368⟩. ⟨hal-01229680⟩

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