Shifted symmetric functions and multirectangular coordinates of Young diagrams

Abstract

In this paper, we study shifted Schur functions $S_{\mu }^{\star }$, as well as a new family of shifted symmetric functions ${\mathfrak{K}}_{\mu }$ 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.

2nd version: minor modifications after referee comments

References in corpus (3)

• Jack polynomials and free cumulants
• A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group
• Asymptotics of characters of symmetric groups, genus expansion and free probability

Cited by in corpus (1)

• The top-degree part in the Matchings-Jack Conjecture