Schur-Positivity in a Square

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 Schurpositivity of a family of symmetric functions. Given a partition ν, we denote by ν c its complement in a square partition (m m ). We conjecture a Schur-positivity criterion for symmetric functions of the form s µ ′ sµc − s ν ′ sνc , where ν is a partition of weight |µ| − 1 contained in µ and the complement of µ is taken in the same square partition as the complement of ν. We prove the conjecture in many cases.

Classifying the near-equality of ribbon Schur functions

“…An especially notorious problem is to classify Schur-positivity for differences of skew Schur functions, which are generalizations of Schur functions. Partial results exist [2,17], but it even remains unknown when two skew Schur functions are equal. Fortunately, more is known in the case of ribbon Schur functions, which are a special case of skew Schur functions that are indexed by compositions.…”