Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition

Maxim GurevichChuijia Wang

Mar 2023

摘要

We employ general parabolic recursion methods to demonstrate the recentlydevised hypercube formula for Kazhdan-Lusztig polynomials of $S_n$, andestablish its generalization to the full setting of a finite Coxeter systemthrough algebraic proof. We introduce procedures for positive decompositions of$q$-derived Kazhdan-Lusztig polynomials within this setting, that utilizeclassical Hecke algebra positivity phenomena of Dyer-Lehrer andGrojnowski-Haiman. This leads to a distinct algorithmic approach to thesubject, based on induction from a parabolic subgroup. We propose suitable weakvariants of the combinatorial invariance conjecture and verify their validityfor permutation groups.

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