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The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals

Grant T. Barkley, Christian Gaetz

TL;DR

This work studies the BBDVW conjecture for Kazhdan--Lusztig polynomials via hypercube decompositions in Bruhat intervals, connecting it to the Combinatorial Invariance Conjecture. It introduces $N_{u,v,I}$, $Q_{u,v,I}$, and the relative $\widetilde{R}$-polynomials $\widetilde{R}_{u,v,I}$, and develops notions of strong hypercube clusters, the numerical criterion, and property (E). By establishing implications between the BBDVW identity and related recurrences under these hypotheses, the authors prove the conjecture for lower intervals $[e,v]$ in the symmetric group, using a refined combinatorial framework (blocks, antichains, camels, caravans). This structural analysis yields a combinatorial pathway to compute $P_{u,v}$ from poset data and strengthens the connection between hypercube decompositions and invariant KL polynomials in a nontrivial interval class.

Abstract

Blundell, Buesing, Davies, Veličković, and Williamson (BBDVW) introduced the notion of a hypercube decomposition of an interval in Bruhat order. They conjectured a recursive formula in terms of this structure which, if shown for all intervals, would imply the Combinatorial Invariance Conjecture of Lusztig and Dyer, for Kazhdan-Lusztig polynomials of the symmetric group. In this article, we prove implications between the BBDVW Conjecture and several other recurrences for hypercube decompositions, under varying hypotheses, which have appeared in the recent literature. As an application, we prove the BBDVW Conjecture for lower intervals $[e,v]$, the first non-trivial class of intervals for which it has been established.

The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals

TL;DR

This work studies the BBDVW conjecture for Kazhdan--Lusztig polynomials via hypercube decompositions in Bruhat intervals, connecting it to the Combinatorial Invariance Conjecture. It introduces , , and the relative -polynomials , and develops notions of strong hypercube clusters, the numerical criterion, and property (E). By establishing implications between the BBDVW identity and related recurrences under these hypotheses, the authors prove the conjecture for lower intervals in the symmetric group, using a refined combinatorial framework (blocks, antichains, camels, caravans). This structural analysis yields a combinatorial pathway to compute from poset data and strengthens the connection between hypercube decompositions and invariant KL polynomials in a nontrivial interval class.

Abstract

Blundell, Buesing, Davies, Veličković, and Williamson (BBDVW) introduced the notion of a hypercube decomposition of an interval in Bruhat order. They conjectured a recursive formula in terms of this structure which, if shown for all intervals, would imply the Combinatorial Invariance Conjecture of Lusztig and Dyer, for Kazhdan-Lusztig polynomials of the symmetric group. In this article, we prove implications between the BBDVW Conjecture and several other recurrences for hypercube decompositions, under varying hypotheses, which have appeared in the recent literature. As an application, we prove the BBDVW Conjecture for lower intervals , the first non-trivial class of intervals for which it has been established.

Paper Structure

This paper contains 17 sections, 30 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kazhdan--Lusztig polynomials
  4. Hypercube decompositions
  5. BBDVW polynomials
  6. $N_{u,v,I}$
  7. $Q_{u,v,I}$
  8. An alternating sum formula
  9. New expressions for the BBDVW polynomials
  10. Strongness, the numerical criterion, and property (E)
  11. The relative $\widetilde{R}$-polynomial
  12. From relative $\widetilde{R}$-polynomials to the BBDVW formula
  13. Property (E)
  14. Hypercube decompositions of lower intervals
  15. Blocks of hypercube decompositions
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $I$ be any hypercube decomposition of a lower interval $[e,v] \subset S_n$. Then

Figures (2)

  • Figure 1: An example of the involution $\iota$ from the proof of \ref{['prop:relative-tR-gen-func']}. The order ideal $I$ is shown in blue. The reflection order is $(1\:2)\prec(1\:3)\prec(2\:3)$. The path $\gamma$ is shown with red arrows, and the marked point is highlighted in pink. In this example, $\gamma$ is increasing at $213$ and decreasing at $123$, so $\iota$ swaps the marked point between those two vertices.
  • Figure 2: An example of a caravan with four blocks (shaded in red) and two camels. All edges are implicitly directed towards the right.

Theorems & Definitions (63)

  • Conjecture 1.1: Lusztig c. 1983; Dyer Dyer1987
  • Conjecture 1.2: Blundell
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 2.1: Kazhdan--Lusztig Kazhdan-Lusztig-1
  • Definition 2.2: Dyer1987
  • ...and 53 more