A Crystal Analysis of $P$-Arrays

TL;DR

This paper constructs a crystal structure on $P$-arrays, providing a combinatorial framework that refines existing $s$-positivity results for chromatic symmetric functions of certain posets.

Contribution

It introduces a new crystal structure on $P$-arrays that enhances understanding of $s$-positivity in chromatic symmetric functions and suggests a generalization of Robinson-Schensted correspondence.

Findings

Crystal components have $s$-positive characters.

Refines Gasharov's $s$-positivity theorems.

Hints at a generalized Robinson-Schensted correspondence.

Abstract

Gasharov introduced the combinatorial objects known as $P$-arrays to prove $s$-positivity for the chromatic symmetric functions of incomparability graphs of (3+1)-free posets. We define a crystal, a directed colored graph with some additional axioms, on the set of $P$-arrays. The components of the crystal have $s$-positive characters, thereby refining the $s$-positivity theorems of Gasharov, as well as Shareshian and Wachs. The crystal hints at a possible generalization of the Robinson-Schensted correspondence applied to $P$-arrays.