Proof of Lassalle's Positivity Conjecture on Schur Functions

TL;DR

This paper proves Lassalle's conjecture that certain evaluations of Schur functions are positive, using Laguerre-Pólya-Schur theory, thus confirming a conjecture related to Catalan numbers and symmetric functions.

Contribution

The paper provides a proof of Lassalle's positivity conjecture for Schur functions using advanced multiplier sequence theory.

Findings

Confirmed Lassalle's conjecture on positivity of Schur function evaluations.

Established a connection between symmetric functions and Laguerre-Pólya-Schur theory.

Validated the positivity property for all partitions under the given homomorphism.

Abstract

In the study of Zeilberger's conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let $(t)_n$ denote the rising factorial, and let $\Lambda_{\mathbb{R}}$ denote the algebra of symmetric functions with real coefficients. If $\varphi$ is the homomorphism from $\Lambda_{\mathbb{R}}$ to $\mathbb{R}$ defined by $\varphi(h_n)={1}/{((t)_nn!)}$ for some $t>0$, then for any Schur function $s_{\lambda}$, the value $\varphi(s_{\lambda})$ is positive. In this paper, we provide an affirmative answer to Lassalle's conjecture by using the Laguerre-P\'olya-Schur theory of multiplier sequences.