Proof of Lassalle's Positivity Conjecture on Schur Functions

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 $\phi$ is the homomorphism from ${\Lambda }_{\mathbb{R}}$ to $\mathbb{R}$ defined by $\phi (h_n)=1/((t{)}_{n}n!)$ for some $t>0$, then for any Schur function $s_{\lambda }$, the value $\phi (s_{\lambda })$ is positive. In this paper, we provide an affirmative answer to Lassalle's conjecture by using the Laguerre-Pólya-Schur theory of multiplier sequences.

8 pages

References in corpus (1)

• A probabilistic interpretation of a sequence related to Narayana polynomials