Necessary conditions for Schur-maximality

TL;DR

This paper confirms conjectures about Schur-maximality for equitable ribbons, establishing conditions under which differences of ribbons are Schur-positive, and identifies minimal equitable ribbons in the Schur-positivity order.

Contribution

It proves conjectures for equitable ribbons forming chains, confirms minimal equitable ribbons, and provides diagrammatic conditions for Schur-positivity of ribbon differences.

Findings

Confirmed conjecture for equitable ribbons forming a chain

Established sufficient conditions for Schur-positivity of ribbon differences

Identified necessary conditions based on row arrangements

Abstract

McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\leq _s A$ if $s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length $a$ or $(a+1)$ for $a\geq 2$. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal. Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length $a$ being at the end, or on rows of length $(a+1)$ being evenly distributed.