Shifted symmetric functions and multirectangular coordinates of Young diagrams

Abstract

In this paper, we study shifted Schur functions $\mathit{S}^{*}_{\mu}$, as well as a new family of shifted symmetric functions $K_{\mu}$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with nonnegative coefficients when written in terms of falling factorials. We then propose a conjectural generalization to the Jack setting. This conjecture is a lifting of Knop and Sahi's positivity result for usual Jack polynomials and resembles recent conjectures of Lassalle. We prove our conjecture for one-part partitions.