Appears in collection : Combinatorics and Arithmetic for Physics - 2024

The kinematic spaces modeling massless particle scattering can be parametrized in different ways such as via the spinor helicity formalism or using momentum twistor coordinates. Either case yields an algebraic variety that we call the spinor helicity variety, respectively the momentum twistor variety. In a general set up (not assuming neither planarity nor dual conformal symmetry in the model) both varieties are isomorphic to (quotients of) partial flag varieties and therefore inherit a cluster structure. I will present results indicating that these cluster structures are relevant in the bootstrap for the amplitude and also exhibit embeddings of the relevant partial flag varieties into Grassmannian respecting the cluster structure. Based on joint works with James Drummond, Ross Glew, respectively with Jianrong Li.