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Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals

Grant T. Barkley, Christian Gaetz

TL;DR

This work develops a strong hypercube decomposition framework to address the Combinatorial Invariance Conjecture for Kazhdan-Lusztig $R$-polynomials. By defining $\widetilde{H}_{u,v,I}(q)$ from a strong hypercube decomposition and analyzing its relation to $\widetilde{R}_{u,v}(q)$ through reflection orders, the authors prove a new instance of invariance for elementary intervals in $S_n$, extending beyond lower intervals. The key technical advance is showing $\widetilde{R}_{u,v}$ equals $\widetilde{H}_{u,v,I}$ for standard decompositions and, in simple intervals, for diamond-closed ideals arising from reflection subgroups. This approach offers a promising combinatorial path toward full invariance, potentially applying to broader Coxeter types given further structural decompositions and isomorphism considerations. The work also relates to parallel conjectures by Brenti and Marietti, framing a recurrence that may someday compute $\widetilde{R}$ purely from the interval’s combinatorial structure.

Abstract

We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig $R$-polynomials in the case of elementary intervals in $S_n$. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.

Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals

TL;DR

This work develops a strong hypercube decomposition framework to address the Combinatorial Invariance Conjecture for Kazhdan-Lusztig -polynomials. By defining from a strong hypercube decomposition and analyzing its relation to through reflection orders, the authors prove a new instance of invariance for elementary intervals in , extending beyond lower intervals. The key technical advance is showing equals for standard decompositions and, in simple intervals, for diamond-closed ideals arising from reflection subgroups. This approach offers a promising combinatorial path toward full invariance, potentially applying to broader Coxeter types given further structural decompositions and isomorphism considerations. The work also relates to parallel conjectures by Brenti and Marietti, framing a recurrence that may someday compute purely from the interval’s combinatorial structure.

Abstract

We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig -polynomials in the case of elementary intervals in . This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.
Paper Structure (10 sections, 18 theorems, 44 equations)

This paper contains 10 sections, 18 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bruhat order
  4. Kazhdan--Lusztig polynomials
  5. Reflection orders and $\widetilde{R}$-polynomials
  6. Strong hypercube decompositions
  7. Diamond-closed order ideals
  8. Standard hypercube decompositions
  9. Extension to other types
  10. The conjecture of Brenti--Marietti

Key Result

Theorem 1.6

Suppose that $[u,v] \subset S_n$ is elementary and $[u,v] \cong [u',v'] \subset S_m$. Then

Theorems & Definitions (56)

  • Conjecture 1.1: Lusztig 1980s; Dyer Dyer1987
  • Conjecture 1.2: Equivalent formulation of Conj. \ref{['conj:cic']}
  • Conjecture 1.3
  • Remark 1.4
  • Definition 1.5
  • Example 1
  • Theorem 1.6
  • Example 2
  • Definition 2.1
  • Remark 2.2
  • ...and 46 more