Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

2015/02/13 by Clearman, Samuel, Hyatt, Matthew, Shelton, Brittany +1 · 4 citations
#05E05 #05E10 #20C08 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1502.04633

Abstract

For irreducible characters \ χ_q^λ | λ\vdash n \, induced sign characters \ ε_q^λ | λ\vdash n \, and induced trivial characters \ η_q^λ | λ\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 χ_q^λ(q^l(w)/2C'_w(q)), ε_q^λ(q^l(w)/2 C'_w(q)), and \smashη_q^λ(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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