Standard complexes of matroids and lattice paths

TL;DR

This paper introduces and analyzes 'standard complexes' associated with matroids, exploring their properties, invariance under duality, and explicit forms for lattice path matroids using combinatorial methods.

Contribution

It defines standard complexes related to matroids, studies their invariance under duality, and explicitly characterizes them for lattice path matroids with combinatorial techniques.

Findings

Standard complexes are invariant under matroid duality.

They satisfy a deletion-contraction recurrence for lexicographic order.

Explicit descriptions of these complexes are provided for lattice path matroids.

Abstract

Motivated by Gr\"obner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.