From amplituhedra to flagtopes

The amplituhedron was introduced by Arkani-Hamed and Trnka (2014) to geometrically encode scattering amplitudes in high-energy physics. It can be viewed as the generalization of a cyclic polytope into the Grassmannian, and is defined to be the projection of the totally nonnegative part of one Grassmannian with respect to a totally negative element of a second Grassmannian. We construct a Lie-theoretic generalization of amplituhedra called 'flag amplituhedra', by replacing the pair of Grassmannians in the definition with an arbitrary pair of partial flag varieties for a reductive group $G$. We study in detail an example of a flag amplituhedron which is not a Grassmannian amplituhedron; it is a curved hexagon in $\mathbb{P}^1\times \mathbb{P}^1$.