Combinatorics of m=1 Grasstopes

by Dmitrii Pavlov (Max Planck Institute for Mathematics in the Sciences)

Wednesday 31 Jan 2024, 15:00 → 16:00 Europe/Berlin

A.1.01/03 - Alps (New)

A Grasstope is the image of the totally nonnegative Grassmannian $\mathrm{Gr}_{\geq 0}(k,n)$ under a linear map $\mathrm{Gr}(k,n)\dashrightarrow \mathrm{Gr}(k,k+m)$. 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. In this talk, I will discuss combinatorics and geometry of Grasstopes in the m=1 case. In particular, I will show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition and argue that amplituhedra are (in a certain sense) minimal Grasstopes. This is joint work with Yelena Mandelshtam and Lizzie Pratt.