Double shortcuts of standard hypercube decompositions

TL;DR

This paper investigates double shortcuts in hypercube decompositions of Bruhat intervals, showing their implications for a conjecture related to Kazhdan--Lusztig polynomials.

Contribution

It establishes that a specific conjecture holds for standard hypercube decompositions, advancing understanding of hypercube structures in algebraic combinatorics.

Findings

The conjecture holds for standard hypercube decompositions.

Implications for the broader Combinatorial Invariance Conjecture.

Potential pathway to prove the conjecture for all hypercube decompositions.

Abstract

In this paper, we study the double shortcuts associated with pairs of standard hypercube decompositions of arbitrary Bruhat intervals in the symmetric group. Our results imply that a conjecture stated in [Bull. London Math. Soc., 57 (2025), no. 8] holds for the class of standard hypercube decompositions. If this conjecture were to hold for all hypercube decompositions, then the Combinatorial Invariance Conjecture for Kazhdan--Lusztig polynomials would follow.