Quiver matroids -- Matroid morphisms, quiver Grassmannians, their Euler characteristics and $\mathbb{F}_1$-points

TL;DR

This paper develops the theory of quiver matroids, introduces their morphisms, and constructs moduli spaces that connect combinatorial structures with algebraic geometry over the field with one element, linking point counts to Euler characteristics.

Contribution

It introduces morphisms for matroids with coefficients and quiver matroids, generalizes to quiver matroid bundles, and constructs their moduli spaces with an $un$-point interpretation.

Findings

Defined morphisms for matroids with coefficients and quiver matroids.

Constructed moduli spaces as $un$-analogues of quiver Grassmannians.

Linked $un$-points of moduli spaces to Euler characteristics of complex quiver Grassmannians.

Abstract

In this paper, we introduce morphisms for matroids with coefficients (in the sense of Baker and Bowler) and quiver matroids. We investigate their basic properties, such as functoriality, duality, minors and cryptomorphic characterizations in terms of vectors, circuits and bases (a.k.a. Grassmann-Pl\"ucker functions). We generalize quiver matroids to quiver matroid bundles and construct their moduli space, which is an $\mathbb{F}_1$-analogue of a complex quiver Grassmannian. Eventually we introduce a suitable interpretation of $\mathbb{F}_1$-points for these moduli spaces, so that in 'nice' cases their number is equal to the Euler characteristic of the associated complex quiver Grassmannian.