Total positivity of Hadamard product of dual Jacobi--Trudi matrices

Abstract

In 1992, Wagner proved that the Hadamard product of two totally positive lower triangular Toeplitz matrices is totally positive. In this work, we strengthen this result by establishing total monomial positivity for the Hadamard product of Jacobi--Trudi matrices. In particular, we resolve a conjecture of Sokal concerning the Hadamard square of Jacobi--Trudi matrices. Moreover, we provide a manifestly positive Schur expansion for the Hadamard square of Jacobi--Trudi matrices indexed by ribbons. In addition, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.

Robert Angarone and Daniel Soskin pointed out a flaw in our paper--specifically, the map $\varphi$ in Theorem 3.2 is not a bijection, so it cannot be a sign-reversing involution. Hence, Sokal's conjecture (Conjecture 1.2) remains open. We would like to thank them for pointing out the error in our paper