Computations with matroids, Grassmannians, and hypersimplices

We use software to study matroid realizations and initial degenerations of Grassmannians. In particular, we use OSCAR to study properties of matroid strata in rank 3 Grassmannians, classifying (3,n) pairs such that all strata are smooth and isolating concrete examples of matroids whose strata are singular. We then use this information, and matroid subdivisions of hypersimplices, to study initial degenerations of Grassmannians. For the (3,8)-Grassmannian, we show all initial degenerations are smooth, using OSCAR to classify all symmetry classes of matroid subdivisions of the (3,8) hypersimplex corresponding to cones in the tropical Grassmannian. Using examples of singular matroid strata, we construct singular initial degenerations of the associated Grassmannian. The content of this talk covers some material from my PhD thesis, and is the result of joint work with Daniel Corey.