An $E_\infty$ structure on the matroid grassmannian

TL;DR

This paper constructs an $E_$-algebra structure on the space of oriented matroids, using polyhedral models and a novel operad, to extend algebraic structures similar to those in $K$-theory.

Contribution

It introduces an $E_$-structure on the matroid Grassmannian, linking polyhedral geometry with operad theory in a new way.

Findings

Established an $E_$-structure on the matroid Grassmannian.

Lifted the Dressian fan structure to a polyhedral model.

Developed a new $E_$ operad from infinite subsets of .

Abstract

In analogy with the origin of the additive structure of $K$-theory, we construct an $E_\infty$ structure on the matroid Grassmannian (the space of oriented matroids) for which the underlying binary operation is the direct sum of matroids. The proof involves lifting the polyhedral fan structure of the Dressian to a polyhedral model for the matroid Grassmannian, and introducing a novel $E_\infty$ operad made from the space of infinite subsets of $\mathbb{N}$.