Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition

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 o...

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Bibliographic Details
Main Authors Gurevich, Maxim, Wang, Chuijia
Format Journal Article
LanguageEnglish
Published 16.03.2023
Subjects
Mathematics - Combinatorics
Mathematics - Representation Theory
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DOI10.48550/arxiv.2303.09251

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Summary: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.
DOI:10.48550/arxiv.2303.09251