Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition

2023/03/16 by Maxim Gurevich, Gurevich, Maxim, Chuijia Wang +1 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2303.09251

Abstract

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of S_n, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of q-derived Kazhdan-Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants of the combinatorial invariance conjecture and verify their validity for permutation groups.

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• Bruhat intervals that are large hypercubes
• Flip Combinatorial Invariance and Weyl groups

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