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Baker and Bowler (2019) showed that the Grassmannian can be defined over a tract, a field-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 to the Lagrangian Grassmannian which is the set of maximal isotropic subspaces in a
By Boege et al. (2019), the Lagrangian Grassmannian is parameterized into the projective space of dimension
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-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-Bowler 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, generalizing Maurer's homotopy theorem for matroids.
We also prove that if a point in the projective space satisfies the
