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On factorization of matrix of Kazhdan-Lusztig polynomials

Aritra Bhattacharya, Ashish Mishra, Shraddha Srivastava

TL;DR

The paper proves that the matrix of Kazhdan–Lusztig polynomials for the Hecke algebra of a Weyl group associated to a symmetrizable Kac–Moody algebra factorizes into a product of $|S|$ matrices with nonnegative coefficients by leveraging hybrid bases $TC^J$ and restriction maps to parabolic subalgebras. It establishes that the change-of-basis coefficients between $TC^J$ and $TC^I$ are nonnegative polynomials in $q$, enabling a chain-factorization of the KL-polynomial matrix along a chain of subsets of $S$. A geometric proof of positivity via perverse sheaves and Braden’s hyperbolic localization is provided, connecting the algebraic positivity to the geometry of flag varieties. The work also identifies parabolic Kazhdan-Lusztig polynomials as special cases of restriction coefficients and provides explicit computations in dihedral and type $A$ settings, illustrating the scope and limits of the positivity results.

Abstract

Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of $\mathcal{H}$ factorizes into a product of $|S|$ many matrices, each of which has entries as polynomials in $q$ with nonnegative coefficients. To achieve this goal, we use hybrid basis $TC^J$ for $J\subseteq S$ of $\mathcal{H}$, defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from $TC^J$-basis to $TC^I$-basis for $I\subset J$. Equivalently, these coefficients can be computed using a natural restriction map from $\mathcal{H}$ to the parabolic Hecke algebra $\mathcal{H}_J$. Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.

On factorization of matrix of Kazhdan-Lusztig polynomials

TL;DR

The paper proves that the matrix of Kazhdan–Lusztig polynomials for the Hecke algebra of a Weyl group associated to a symmetrizable Kac–Moody algebra factorizes into a product of matrices with nonnegative coefficients by leveraging hybrid bases and restriction maps to parabolic subalgebras. It establishes that the change-of-basis coefficients between and are nonnegative polynomials in , enabling a chain-factorization of the KL-polynomial matrix along a chain of subsets of . A geometric proof of positivity via perverse sheaves and Braden’s hyperbolic localization is provided, connecting the algebraic positivity to the geometry of flag varieties. The work also identifies parabolic Kazhdan-Lusztig polynomials as special cases of restriction coefficients and provides explicit computations in dihedral and type settings, illustrating the scope and limits of the positivity results.

Abstract

Let be the Hecke algebra of the Coxeter system over , where is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of factorizes into a product of many matrices, each of which has entries as polynomials in with nonnegative coefficients. To achieve this goal, we use hybrid basis for of , defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from -basis to -basis for . Equivalently, these coefficients can be computed using a natural restriction map from to the parabolic Hecke algebra . Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.
Paper Structure (12 sections, 27 theorems, 112 equations)

This paper contains 12 sections, 27 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Hecke algebras
  3. The restriction map
  4. The hybrid basis
  5. Proofs of main results
  6. Parabolic Kazhdan-Lusztig polynomials
  7. Examples of restriction coefficients
  8. The elements $R_w$
  9. Dihedral groups
  10. Parabolic Kazhdan-Lusztig polynomials for type $A$
  11. Restriction coefficients for type A
  12. A geometric proof of positivity of restriction coefficients

Key Result

Theorem A

For $u \in W^J$ and $w \in W$, the coefficients of $(T_{u^{-1}}C_w)|_J$ with respect to the Kazhdan-Lusztig basis of $\mathcal{H}_J$ are polynomials in $q$ with nonnegative coefficients.

Theorems & Definitions (53)

  • Theorem A
  • Theorem B
  • Theorem C
  • Example 2.1
  • Proposition 2.2: Proposition 4.4 Casselmannnotes
  • Proposition 2.3: Corollary 4.5 of Casselmannnotes
  • Proposition 2.4: Proposition 5.2 of Casselmannnotes
  • Proposition 3.1
  • proof
  • Example 3.2
  • ...and 43 more