Baker--Bowler theory for Lagrangian Grassmannians

TL;DR

The paper generalizes Baker--Bowler matroid theory to the Lagrangian Grassmannian, viewing maximal isotropic subspaces through restricted Grassmann--Plücker relations that induce antisymmetric matroids. It provides two cryptomorphic perspectives—basis- and circuit-based—for antisymmetric matroids, and extends representability and minor-closure concepts to tracts, partial fields, and hyperfields. A homotopy theorem for the associated transversal basis graphs underpins a robust Tutte-type theorem for the Lagrangian case, linking classical matroid theory with type-C combinatorics. The work also connects to delta-matroids, symmetric/even symmetric matroids, gaussoids, and tropical geometry via the symplectic Dressian and isotropic tropical linear spaces, offering a unified framework for Lagrangian Grassmannians over diverse coefficient systems. Overall, the results provide a comprehensive, tract-compatible atlas for Lagrangian-type combinatorics and their geometric realizations.

Abstract

Baker and Bowler showed that the Grassmannian can be defined over a tract, a field-like structure generalizing both partial fields and hyperfields. This notion unifies theories of matroids over partial fields, valuated matroids, and oriented matroids. We extend Baker--Bowler theory to the Lagrangian Grassmannian which is the set of maximal isotropic subspaces in a $2n$-dimensional symplectic vector space. By Boege et al., the Lagrangian Grassmannian is parameterized as a subset of the projective space of dimension $2^{n-2}(4+\binom{n}{2})-1$ and its image is cut out by certain quadrics. We simplify a list of quadrics so that these are apparently induced by the Laplace expansions only concerning principal and almost-principal minors of a symmetric matrix. From the idea that the strong basis exchange axiom of matroids captures the combinatorial essence of the Grassmann--Plücker relations, we define matroid-like objects, called antisymmetric matroids, derived from the quadrics for the Lagrangian Grassmannian. We also provide a cryptomorphic definition in terms of circuits capturing the orthogonality and maximality of a Lagrangian subspace. We define antisymmetric matroids over tracts in two equivalent ways, which generalize both BB theory and the parameterization of the Lagrangian Grassmannian. It provides a new perspective on the Lagrangian Grassmannian over hyperfields such as the tropical hyperfield and the sign hyperfield. Our proof involves a homotopy theorem for graphs associated with antisymmetric matroids, which generalizes Maurer's homotopy theorem for matroids. We also prove that if a point in the projective space satisfies the $3$-/$4$-term quadratic relations for the Lagrangian Grassmannian and its supports form the bases of an antisymmetric matroid, then it satisfies all quadratic relations, a result motivated by the earlier work of Tutte for matroids and the Grassmannian.

Figures (6)

• Figure 1: The basis polytope of a symmetric matroid $M=([3]\cup[3]^*,\{1^*2^*3^*, 12^*3^*, 1^*23^*, 1^*2^*3, 123\})$. The six mid-points $\pm \mathbf{e}_i$ on $1$-dimensional faces represent almost-transversals of the antisymmetric matroid $N$ such that $\mathcal{B}(M) = \mathcal{B}(N) \cap \mathcal{T}_n$.
• Figure 2: Descriptions of Case I in the proof of Theorem \ref{['thm: homotopy']}. Solid lines represent edges of weight $1$, double lines represent edges of weight $2$, and dashed lines represent paths in $\mathcal{G}_M$.
• Figure 3: Descriptions of Case II.1 in the proof of Theorem \ref{['thm: homotopy']}.
• Figure 4: Descriptions of Case II.2 in the proof of Theorem \ref{['thm: homotopy']}.
• Figure 5: Two descriptions of the cycle $C$ of weight $6$ in $\mathcal{G}_M$ in the proof of Lemma \ref{['lem: gamma 6cycle']}. The cycle $C$ can be identified with a $6$-cycle in $G_M$, where solid dots represent transversal bases and hollow dots represent almost-transversal bases of $M$.

Theorems & Definitions (148)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: see Oxley2011matroid
• Lemma 2.2: Minty's Painting Axiom Minty1966
• Lemma 2.3: folklore
• proof
• proof : Proof of Theorem \ref{['thm: parameterization']}
• Definition 3.1: Antisymmetric Matroids
• Lemma 3.2