Necessary Conditions for Schur-Positivity

TL;DR

This paper establishes necessary conditions for the difference of skew Schur functions to be Schur-positive, advancing understanding of the structural properties influencing Schur-positivity in algebraic combinatorics.

Contribution

It provides the first set of necessary conditions for Schur-positivity of skew Schur function differences, extending previous results on equality conditions.

Findings

Identifies necessary conditions for Schur-positivity of skew Schur function differences.

Strengthens previous results on conditions for s_A = s_B.

Provides insights into the structural properties affecting Schur-positivity.

Abstract

In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s_A - s_B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s_A = s_B, and we deduce a strengthening of their result as a special case.