Geodesic
Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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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. 
@article{DMTCS_2013_special_264_a52,
     author = {Clearman, Sam and Hyatt, Matthew and Shelton, Brittany and Skandera, Mark},
     title = {Evaluations of {Hecke} algebra traces at {Kazhdan-Lusztig} basis elements},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.2368},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2368/}
}
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Clearman, Sam; Hyatt, Matthew; Shelton, Brittany; Skandera, Mark. Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.2368. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2368/

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