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Schur-positivity in a Square

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 λ, we denote by λ^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{μ'}s_{μ^c}-s_{λ'}s_{λ^c}, where λis a partition of weight |μ|-1 contained in μand the complement of μis taken in the same square partition as the complement of λ. We prove the conjecture in many cases.