Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

TL;DR

This paper establishes new combinatorial and algebraic relations on skew Schur functions that prove certain differences are Schur positive, confirming conjectures and extending positivity results to Schubert classes.

Contribution

It introduces novel relations on skew Schur functions using jeu de taquin and Jacobi-Trudi determinants, proving Schur positivity of specific differences and applying these to ribbons and Schubert classes.

Findings

Certain skew Schur function differences are Schur positive.

Confirmed McNamara's Schur positivity conjecture for ribbon basis.

Demonstrated Schubert positivity of specific class differences.

Abstract

Some new relations on skew Schur function differences are established both combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive.