Proof of Lassalle's Positivity Conjecture on Schur Functions

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.