A note on Combinatorial Invariance of Kazhdan--Lusztig polynomials

TL;DR

This paper introduces new combinatorial concepts like hypercube decomposition and double shortcuts to formulate a conjecture supporting the combinatorial invariance of Kazhdan--Lusztig polynomials for symmetric groups, with related proofs for special cases.

Contribution

It proposes a new combinatorial conjecture based on hypercube decompositions, advancing understanding of Kazhdan--Lusztig polynomial invariance.

Findings

Formulated a conjecture implying combinatorial invariance

Introduced hypercube decomposition and double shortcuts

Proved an analogous conjecture for co-elementary intervals

Abstract

We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the symmetric group. This conjecture has the advantage of being combinatorial in nature. The appendix by Grant T. Barkley and Christian Gaetz discusses the related notion of double hypercubes and proves an analogous conjecture for these in the case of co-elementary intervals.