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Cristina Ballantine
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Rosa Orellana
Keywords:
Schur-positivity, Littlewood-Richardson Coefficients, Kronecker product
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 $\nu$, we denote by $\nu^c$ its complement in a square partition $(m^m)$. We conjecture a Schur-positivity criterion for symmetric functions of the form $s_{\mu'}s_{\mu^c}-s_{\nu'}s_{\nu^c}$, where $\nu$ 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 $\nu$. We prove the conjecture in many cases.