Hadamard Products of dual Jacobi-Trudi matrices

TL;DR

This paper investigates the positivity properties of Hadamard products of Jacobi-Trudi matrices, confirming Sokal's conjecture for ribbon-shaped minors and providing new Schur positivity results with representation-theoretic insights.

Contribution

It proves Sokal's conjecture for ribbon-like shapes and establishes Schur positivity of Temperley-Lieb immanants on Hadamard products of Jacobi-Trudi matrices.

Findings

Temperley-Lieb immanants are Schur positive for ribbon-like shapes.

Confirmed Sokal's conjecture for minors with ribbon shapes.

Provided a positive Schur expansion and a representation-theoretic proof.

Abstract

We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Mal\'{o} proved that the Hadamard (entrywise) product of two totally positive upper-triangular Toeplitz matrices whose Toeplitz sequences are the coefficient sequences of real-rooted polynomials with nonpositive zeros is again totally positive. Sokal conjectured that this result can be strengthened to total monomial positivity for the Hadamard product of Jacobi-Trudi matrices. In this paper we show that Temperley-Lieb immanants are Schur positive for Hadamard products of Jacobi-Trudi matrices given by ribbon-like skew shapes. In particular, we affirm Sokal's conjecture for minors given by ribbon-like skew shapes. Moreover, we provide a manifestly positive Schur expansion for Temperley-Lieb immanants evaluated on the Hadamard product of Jacobi-Trudi matrices indexed by ribbons. In addition, for the ribbon case, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.