Matroid complexes and Orlik-Solomon algebras

TL;DR

This paper introduces a combinatorial differential graded model for Orlik-Solomon algebras of supersolvable matroids, extending known algebraic structures and proving their Koszul property using a new approach.

Contribution

It constructs a novel combinatorial model for Orlik-Solomon algebras of supersolvable matroids and demonstrates their Koszulness through this framework.

Findings

Constructed a quasi-free differential graded model for the algebra

Established a cooperadic structure in the model

Provided a new proof of Koszul property for supersolvable matroids

Abstract

In this article we construct a combinatorial quasi-free differential graded model for the Orlik-Solomon algebra of supersolvable matroids, which generalizes in a matroidal setting the cdga of admissible graphs introduced by M. Kontsevich for the braid arrangements. Our construction draws on well-known concepts from matroid theory, including modularity, single-element extensions, and generalized parallel connections. We also show that this model carries a cooperadic structure in a suitably generalized sense. As an application, we use this model to give a new proof that the Orlik-Solomon algebras of supersolvable matroids are Koszul.