Baker-Bowler theory for Lagrangian Grassmannians

Abstract: Baker and Bowler (2019) showed that the Grassmannian can be defined over a tract, a ring-like structure generalizing both partial fields and hyperfields. This notion unifies theories for matroids over partial fields, valuated matroids, and oriented matroids. We extend Baker-Bowler theory for the Lagrangian Grassmannian which is the set of maximal isotropic subspaces of a $2n$-dimensional symplectic vector space. By Boege et al. (2019), the Lagrangian Grassmannian is parameterized into the projective space of dimension $2^{n-2}(4+\binom{n}{2})-1$ and its image is exactly the solutions of quadrics induced by determinantal identities of 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 Pl\"{u}cker relations, we define matroid-like objects, called antisymmetric matroids, from the quadrics for the Lagrangian Grassmannian. We also provide its cryptomorphic definition in terms of circuits capturing the orthogonality and maximality of vectors in a Lagrangian subspace. We define antisymmetric matroids over tracts in two equivalent ways, which generalize both Baker-Bolwer theory and the parameterization of the Lagrangian Grassmannian. It provides a new perspective on the Lagrangian Grassmannian over hyperfields, especially, the tropical Lagrangian Grassmannian. Our proof involves a homotopy theorem for graphs associated with antisymmetric matroids, generalizing Maurer's homotopy theorem for matroids. We also prove that if a point in the projective space satisfies $3$-/$4$-term quadrics for the Lagrangian Grassmannian and its supports form the bases of an antisymmetric matroid, then it satisfies all quadrics, which is motivated by the earlier work of Tutte (1958) for matroids and linear spaces.