Noncommutative Schur functions for posets

TL;DR

This paper extends noncommutative Schur function theory to posets, proving Schur positivity for functions linked to P-Knuth equivalence graphs, thereby resolving a conjecture and refining prior results.

Contribution

It develops the theory to prove Schur positivity for P-Knuth equivalence graph functions, settling a conjecture and improving previous findings.

Findings

Proves Schur positivity for P-Knuth equivalence graph functions.

Resolves a conjecture of Kim and the third author.

Refines results on chromatic symmetric functions' Schur positivity.

Abstract

The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any $P$-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.