# Shifted symmetric functions and multirectangular coordinates of Young diagrams

Alexandersson, Per; Féray, Valentin (2017). Shifted symmetric functions and multirectangular coordinates of Young diagrams. Journal of Algebra, 483:262-305.

## 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.

## 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.

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Algebra and Number Theory Algebra and Number Theory English August 2017 22 May 2017 14:58 26 Jan 2022 12:57 Elsevier 0021-8693 Closed Publisher DOI. An embargo period may apply. https://doi.org/10.1016/j.jalgebra.2017.03.036 : FunderSNSF: Grant ID200021_149461: Project TitleDual combinatorics of Jack polynomials

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