Robinson-Schensted correspondence for unit interval orders

TL;DR

This paper introduces a new Robinson-Schensted type correspondence for unit interval orders, conjectures Schur-positivity of associated classes, and proves these conjectures for certain cases, advancing understanding of the Stanley-Stembridge conjecture.

Contribution

It defines e0 la Knuth relations for each unit interval order and proves conjectures relating to Schur-positivity and e0 la Robinson-Schensted insertion for specific classes.

Findings

Conjectured Schur-positivity of e0 la Knuth classes for unit interval orders.

Proved conjectures for e0 la Robinson-Schensted insertion in certain cases.

Connected the framework to D graphs studied by Assaf.

Abstract

The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order $\mathcal P$. In this paper, we define relations \`a la Knuth on the symmetric group for each $\mathcal P$ and conjecture that the associated $\mathcal P$-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, and Guay-Paquet. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of $\mathcal P$-tableaux that occur in the equivalence class. We prove these conjectures for $\mathcal P$ avoiding two specific suborders by introducing $\mathcal P$-analog of Robinson-Schensted insertion, giving an answer to a long standing question of Chow.