Tropical matroid Schubert varieties and the graded M\"obius algebra

TL;DR

This paper introduces tropical matroid Schubert varieties, establishing their cohomology rings as isomorphic to the graded Möbius algebra, thus providing a geometric model for these algebraic structures across all matroids.

Contribution

It extends the geometric framework of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones, via tropical geometry.

Findings

The tropical cohomology ring of the tropical matroid Schubert variety is isomorphic to the graded Möbius algebra.

Provides a geometric model for the Möbius algebra for all matroids.

Extends the geometric setting to non-realisable matroids.

Abstract

We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid $M$ is isomorphic to the graded M\"obius algebra $\operatorname{B}^\bullet(M)$. This yields a geometric model for $\operatorname{B}^\bullet(M)$, extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.