Matroids, Feynman categories, and Koszul duality

TL;DR

This paper introduces operad-like structures for matroid invariants within a Feynman category and establishes a Koszul duality between Chow rings and Orlik--Solomon algebras, generalizing known results.

Contribution

It constructs operads over a new Feynman category for matroid invariants and proves a broad Koszul duality, extending previous work by Getzler.

Findings

Operad structures for matroid invariants are developed.

A Koszul duality between Chow rings and Orlik--Solomon algebras is established.

Provides new interpretations of combinatorial Leray models.

Abstract

We show that various combinatorial invariants of matroids such as Chow rings and Orlik--Solomon algebras may be assembled into "operad-like" structures. Specifically, one obtains several operads over a certain Feynman category which we introduce and study in detail. In addition, we establish a Koszul-type duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a celebrated result of Getzler. This provides a new interpretation of combinatorial Leray models of Orlik--Solomon algebras.