The virtual Euler characteristic for binary matroids

TL;DR

This work defines a virtual Euler characteristic for finite sets of matroid isomorphism classes and computes it explicitly for simple binary matroids, obtaining $\chi(\mathcal{B}(r)) = \prod_{i=1}^r \frac{1}{1-2^i}$. The authors connect matroid counting to point counts on the distinct-column locus of Grassmannians, proving $\sum_{n\ge r} \frac{(-1)^n}{n!} |\mathsf{Gr}^{\mathsf{dc}}(r,n;\mathbb{F}_q)| = \prod_{i=1}^r \frac{1}{1-q^i}$ for any prime power $q$, via induction and combinatorial identities. They further promote these identities to the Grothendieck ring of varieties, showing a recursion that expresses $[\mathsf{Gr}^{\mathsf{dc}}(r+1,n;F)]$ as a sum over lower-rank Grassmannians times configuration spaces, using a stratification into $Z_k$ and $Y_k$ and a Zariski-fibration argument. The results bridge matroid enumeration, finite-field geometry, and motivic invariants, and open directions include extensions to matroids realizable over other finite fields, connections to matroid homology, and beta-invariants.

Abstract

Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of matroids of rank $r$ realizable over $\mathbb{F}_2$ (i.e., binary matroids). We prove this formula by relating the virtual Euler characteristic for binary matroids to the point counts of certain subsets of Grassmanians over finite fields. We conclude by providing several follow-up questions in relation to matroids realizable over other finite prime fields, matroid homology, and beta invariants.

Figures (1)

• Figure 1: Rank 3 simple binary matroids

Theorems & Definitions (17)

• Theorem 1
• Example 1.1
• Theorem 2
• Theorem 3
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3