Newton polytopes of immanants of some combinatorial matrices

TL;DR

This paper investigates the Newton polytopes of immanants for Jacobi-Trudi and Giambelli matrices, verifying saturation properties and explicitly determining key coefficients using lattice path methods.

Contribution

It proves the saturation property for all immanants of Giambelli matrices and verifies it in special cases for Jacobi-Trudi matrices, advancing understanding of their geometric properties.

Findings

Saturation property holds for all immanants of Giambelli matrices.

Saturation property verified in special cases for Jacobi-Trudi matrices.

Explicit coefficients of the largest monomials obtained for Giambelli matrices.

Abstract

The immanants of combinatorial matrices have many significant properties, including m-positivity and Schur positivity. While the immanants of Jacobi-Trudi matrices are known to be both m-positive and Schur positive, those of Giambelli matrices have only been proven to be m-positive, with Schur positivity remaining a conjecture. These positivity properties rely heavily on lattice path interpretations. In this paper, we study the Newton polytopes of immanants for these two classes of matrices. Using the lattice path method, we verify the saturation property for the Newton polytopes of Jacobi-Trudi matrices in special cases. For Giambelli matrices, we prove that this property holds for all immanants. To achieve this, we obtain the explicit coefficients of the largest monomial (in the dominance order) in the monomial expansion of the immanants of Giambelli matrices.