The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals

TL;DR

This paper proves the BBDVW Conjecture for lower intervals in Bruhat order, establishing a key recursive structure that supports the broader invariance conjecture for Kazhdan-Lusztig polynomials in symmetric groups.

Contribution

It demonstrates the BBDVW Conjecture for a significant class of intervals, linking it to other known recurrences and advancing understanding of Kazhdan-Lusztig polynomial invariance.

Findings

Proved BBDVW Conjecture for lower intervals [e,v]

Established implications between various hypercube decomposition conjectures

Connected the conjecture to the broader invariance conjecture in representation theory

Abstract

Blundell, Buesing, Davies, Veli\v{c}kovi\'c, 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.