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Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials

Francesco Esposito, Mario Marietti

Abstract

In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan--Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of $q^h$ of the Kazhdan--Lusztig $\widetilde{R}$-polynomials, for $h\leq 6$. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.

Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials

Abstract

In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan--Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of of the Kazhdan--Lusztig -polynomials, for . This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.
Paper Structure (13 sections, 35 theorems, 30 equations, 3 figures)

This paper contains 13 sections, 35 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-support graphs and flipclasses
  4. Flipclasses and number of increasing paths
  5. Some results on flipclasses
  6. From infinite to finite
  7. Combinatorial invariance of the coefficients of $\widetilde{R}$-polynomials
  8. $h=3$
  9. $h=4$
  10. $h=5$
  11. $h=6$
  12. Consequences
  13. A conjecture

Key Result

Theorem 2.1

Let $u,v\in \mathfrak S_n$. The isomorphism class (as a poset) of the Bruhat interval $[u,v]$ determines the isomorphism class (as a directed graph) of the graph $B([u,v])$.

Figures (3)

  • Figure 1: A segment, a diamond, and a $k$-crown
  • Figure 2: Support and time-support graphs
  • Figure 3: The time-support graph of a flipclass with $t$-vector $(1,3,4,3,1)$

Theorems & Definitions (86)

  • Conjecture 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 76 more