Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

We continue the study of positive geometries underlying the {\it Grassmannian string integrals}, which are a class of "stringy canonical forms", or stringy integrals, over the positive Grassmannian mod torus action, $G_+(k,n)/T$. The leading order of any such stringy integral is given by the canonical function of a polytope, which can be obtained using the Minkowski sum of the Newton polytopes for the regulators of the integral, or equivalently given by the so-called scattering-equation map. The canonical function of the polytopes for Grassmannian string integrals, or the volume of their dual polytopes, is also known as the generalized bi-adjoint $\phi^3$ amplitudes. We compute all the linear functions for the facets which cut out the polytope for all cases up to $n=9$, with up to k=4 and their parity conjugate cases. The main novelty of our computation is that we present these facets in a manifestly gauge-invariant and cyclic way, and identify the boundary configurations of $G_+(k,n)/T$ corresponding to these facets, which have nice geometric interpretations in terms of $n$ points in $(k{-}1)$-dimensional space. All the facets and configurations we discovered up to $n=9$ directly generalize to all $n$, although new types are still needed for higher $n$.