Understand
In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the kinematic space $\mathcal{K}_{k,n}$.
- It was shown there for $m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of a polyhedral complex known as the (nonnegative) tropical Grassmannian.
- In this note, we derive a remarkable planar basis for the space of generalized kinematic invariants which coincides in the case $k=2$ with usual standard planar multi-particle basis for the kinematic space.
- We implement in Mathematica the action on formal linear combinations of planar matroid subdivisions of a boundary operator which, together with the planar basis, determines compatibility for any given poles appearing in the expansion of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$, by computing a certain combinatorial non-crossing condition on the second hypersimplicial faces $\Delta_{2,n-(k-2)}$ of $\Delta_{k,n}$.