Stiefel tropical linear spaces

TL;DR

This paper introduces Stiefel tropical linear spaces, explores their duality with matroid subdivisions, and connects their combinatorics to tropical hyperplane arrangements and the secondary fan of Newton polytopes.

Contribution

It establishes the duality of Stiefel tropical linear spaces with matroid subdivisions and links their combinatorics to tropical hyperplane arrangements and Newton polytope fans.

Findings

Stiefel tropical linear spaces are dual to matroid subdivisions of transversal matroid polytopes.

They are related to tropical hyperplane arrangements.

Connections are made with the secondary fan of the Newton polytope.

Abstract

The tropical Stiefel map associates to a tropical matrix A its tropical Pluecker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are dual to certain matroid subdivisions of polytopes of transversal matroids, and we relate their combinatorics to a canonically associated tropical hyperplane arrangement. We also explore a broad connection with the secondary fan of the Newton polytope of the product of all maximal minors of a matrix. In addition, we investigate the natural parametrization of L(A) arising from the tropical linear map defined by A.