On local Dressians of matroids

TL;DR

This paper explores the fan structure of Dressians associated with matroids, revealing their relation to matroid polytopes and establishing properties of indecomposable matroids, including binary and non-binary cases.

Contribution

It demonstrates that the fan structure on local Dressians matches the secondary fan of the matroid polytope and characterizes indecomposable matroids, including binary and non-binary examples.

Findings

Fan structure on $ ext{Dr}( extbf{M})$ coincides with the secondary fan of the matroid polytope.

Matroid subdivision is determined by its 3-dimensional skeleton.

Binary matroids are indecomposable; a non-binary indecomposable matroid is provided.

Abstract

We study the fan structure of Dressians $\Dr(d,n)$ and local Dressians $\Dr(\cM)$ for a given matroid $\cM$. In particular we show that the fan structure on $\Dr(\cM)$ given by the three term Pl\"ucker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope $P(\cM)$. As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse.