Positivity results on ribbon Schur function differences

TL;DR

This paper investigates when differences of skew Schur functions are Schur positive, focusing on ribbons, and characterizes the poset structure for multiplicity-free ribbons as a product of two chains.

Contribution

It provides necessary and sufficient conditions for multiplicity-free ribbons to be Schur positive and describes the poset structure as a product of two chains.

Findings

Positivity conditions for multiplicity-free ribbons identified.

The poset of such ribbons is a product of two chains.

Advances understanding of Schur positivity in skew Schur functions.

Abstract

There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur-positivity is essentially a product of two chains.