Multi-splits and tropical linear spaces from nested matroids

TL;DR

This paper provides a combinatorial description of multi-splits of hypersimplices, relates them to nested matroids, and offers bounds on the facets of secondary polytopes, advancing understanding of tropical linear spaces.

Contribution

It introduces a new combinatorial framework linking multi-splits of hypersimplices to nested matroids and describes all multi-splits of products of simplices.

Findings

Explicit description of facets of secondary polytopes of hypersimplices.

Relation established between multi-splits and nested matroids.

Lower bounds on the number of facets of secondary polytopes.

Abstract

In this paper we present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show a relation between the cells in a multi-split of the hypersimplex and nested matroids. Moreover, we get a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.