Schur-positivity in a Square

TL;DR

This paper investigates Schur-positivity of certain symmetric functions related to square partitions, proposing a conjecture and proving it in many cases to advance understanding of Schur-positivity criteria.

Contribution

It introduces a conjecture on Schur-positivity for functions involving complements in square partitions and proves it in numerous cases.

Findings

Conjectured a Schur-positivity criterion for specific symmetric functions.

Proved the conjecture in many cases.

Enhanced understanding of Schur-positivity in square partitions.

Abstract

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by \lambda^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{\mu'}s_{\mu^c}-s_{\lambda'}s_{\lambda^c}, where \lambda is a partition of weight |\mu|-1 contained in \mu and the complement of \mu is taken in the same square partition as the complement of \lambda. We prove the conjecture in many cases.