Combinatorial Theory

A Grasstope is the image of the totally nonnegative Grassmannian \(\operatorname{Gr}_{\geq 0}(k,n)\) under a linear map \(\operatorname{Gr}(k,n)\dashrightarrow \operatorname{Gr}(k,k+m)\). This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. We study Grasstopes in the \(m=1\) case and show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition, extending the work of Karp and Williams. Inspired by this characterization, we also suggest a notion of a Grasstope arising from an arbitrary oriented matroid.

Mathematics Subject Classifications: 05E14, 14N10, 14M15

Keywords: Grasstope, Grassmannian, amplituhedron, hyperplane arrangements, sign vectors, oriented matroids