Tropical symplectic Grassmannian

Given a 2n-dimensional vector space V with a symplectic form w, its linear subspace L is called isotropic if all vectors in L are pairwise orthogonal with respect to the form w. The symplectic Grassmannian SpGr(k,2n) is the space of all k-dimensional isotropic linear subspaces of V. We formulate tropical analogues of several equivalent characterizations of this space. These tropical analogues are not equivalent in general; we give all implications between them, and some counter examples. In the case k=2, we study the fan structure of the respective tropical symplectic Grassmannian, giving a count of the number of rays and maximal cones.

This is based on joint work with Jorge Alberto Olarte.