Schur-positivity in a square (Q743661)

Summary: 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 \(s_{\mu^\prime}s_{\mu^c}-s_{\nu^\prime}s_{\nu^c}\), 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.