New light on Bergman complexes by decomposing matroid types

TL;DR

This paper introduces a general and optimal method for decomposing matroid types within Bergman complexes into direct summands, simplifying their structure and providing a unified framework for various subdivisions.

Contribution

It presents a new, more general proof of the subdivision of Bergman complexes and introduces a finest possible direct sum decomposition for matroid faces.

Findings

Decomposition into direct sums applies to all faces of Bergman complexes.

The method generalizes previous formulas for special cases.

The decomposition is proven to be the finest possible.

Abstract

Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas proposed by Ardila/Klivans and Feichtner/Sturmfels for special cases, we present a decomposition into direct sums working for faces of any of these complexes. Additionally we show that it is the finest possible decomposition for faces of the Bergman complex.