Hadamard Products of dual Jacobi-Trudi matrices

TL;DR

This work addresses when Hadamard (entrywise) products of Jacobi–Trudi minors preserve positivity, focusing on the Temperley–Lieb (TL) immanants. It proves Schur-positivity for TL-immanants of Hadamard products indexed by ribbon-like, $3\times 2$-avoiding skew shapes, and provides an explicit manifestly positive Schur expansion in the ribbon case, together with a representation-theoretic model for the ribbon expansions. The approach leverages lattice-path formulations, TL algebra, and $\mathfrak{S}_n\times\mathfrak{S}_n$ representation theory to interpret coefficients combinatorially and algebraically, yielding a pathway toward Sokal’s conjecture in this restricted setting. The results explain why the sign-reversing involution techniques extend in the ribbon/$3\times 2$-avoiding regime but face obstructions beyond it, and they illuminate a representation-theoretic realization of the Schur expansions via poset topology. Altogether, the paper advances our understanding of monomial/Schur positivity under Hadamard products and connects total positivity to rich combinatorial and representation-theoretic structures.

Abstract

We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Maló 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.

Figures (8)

• Figure 1: A lattice path $p:A \to B$ from $A=(6,5)$ to $B=(3,0)$ with $\operatorname{wt}_{\mathbf{x}}(p) = x_5x_4x_2$.
• Figure 2: Multiplicative generators for $\operatorname{TL}_4(2)$.
• Figure 3: Concatenation of Kauffman diagrams for the product $t_1t_2t_1t_1t_3=2t_1t_3$.
• Figure 4: On the left, a $((2,2,2,2), \emptyset)$-subnetwork $H$. On the right, the four path families that cover $H$, with different colors indicating distinct paths. Below each path family, its permutation type is displayed. Summing over all types, we obtain $\beta(H) = 1 + \mathfrak{s}_1 + \mathfrak{s}_2 + \mathfrak{s}_1\mathfrak{s}_2 = (1+\mathfrak{s}_1)(1+\mathfrak{s}_2)$.
• Figure 5: Using the subnetwork $H$ from \ref{['fig:subnetwork and beta']}, we implement the visual method for obtaining $\psi(H)$ and $\epsilon(H)$.

Theorems & Definitions (51)

• Conjecture 1.1: Sokal's conjecture
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Example 2.2
• Lemma 2.3
• Definition 2.4
• Corollary 2.5
• Lemma 3.1