Shifted symmetric functions and multirectangular coordinates of Young diagrams

TL;DR

This paper explores shifted symmetric functions related to Young diagrams, demonstrating their polynomial nature in multirectangular coordinates and proposing a conjecture extending these properties to Jack polynomials.

Contribution

It introduces new shifted symmetric functions linked to Kostka numbers and proves their polynomiality, also proposing a conjecture for Jack polynomials with partial proof.

Findings

Shifted Schur functions are polynomials in multirectangular coordinates.

New shifted symmetric functions $rak{K}_ u$ are introduced and shown to have nonnegative coefficients.

Conjecture for Jack polynomials is proposed and proved for one-part partitions.

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.