Tropicalizing Principal Minors of Positive Definite Matrices

TL;DR

This paper explores the tropicalization of principal minors of positive definite matrices, revealing geometric structures and inequalities that deepen understanding of matrix properties through tropical geometry.

Contribution

It characterizes the tropicalization of positive definite matrices' principal minors as a polyhedral set intersecting tropical flag varieties, introducing new algebraic inequalities.

Findings

Tropicalization coincides with intersection of tropical flag variety and submodular cone.

Cells in the subdivision relate to realizable matroids.

New inequalities among principal minors are discovered.

Abstract

We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of $M$-concave functions on the discrete $n$-dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive semidefinite matrices of a fixed size. We also extend our results to positive semidefinite matrices via taking closures in the tropical semifield $\mathbb{R}\cup\{-\infty\}$.