A Grassmann algebra for matroids

TL;DR

This paper develops a tropical analogue of Grassmann algebra to better understand tropical linear spaces and valuated matroids, establishing new cryptomorphic characterizations.

Contribution

It introduces an idempotent exterior algebra framework that parallels classical Grassmann algebra, providing new cryptomorphisms for valuated matroids.

Findings

Tropical linear spaces are characterized via wedge powers and Plucker vectors.

A tensor satisfies tropical Plucker relations iff certain wedge powers are rank-one free modules.

The framework unifies classical and tropical matroid theories through algebraic formalism.

Abstract

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.