Proof of a Positivity Conjecture on Schur Functions
William Y.C. Chen, Anne X.Y. Ren and Arthur L.B. Yang
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 Λ_{R} denote the algebra of symmetric functions with real coefficients. If is the homomorphism from Λ_{R} to R defined by (h_{n}) = 1/((t)_{n}n!) for some t > 0, then for any Schur function s_{λ}, the value (s_{λ}) is positive. In this paper, we provide an affirmative answer to Lassalle's conjecture by using the Laguerre-Pólya-Schur theory of multiplier sequences. AMS Classification: Primary 05E05; Secondary 26C10 Keywords: symmetric function, Schur function, multiplier sequence, totally positive sequence Download: pdf |