Consider a configuration of n labeled points in a Euclidean space. Any linear functional gives an ordering of these points: an ordered partition that we call a sweep, because we can imagine its parts as the sets of points successively hit by a sweeping hyperplane. The set of all such sweeps forms a poset which is isomorphic to a polytope, called the sweep polytope.
I will present several constructions of the sweep polytope, related to zonotopes, projections of permutahedra, monotone path polytopes and sums of k-set polytopes.
Then, instead of recording the whole ordering of the n points, we are interested in the ordering of the first r points we hit, that we call a lineup. We also obtain the structure of a polytope, called the lineup polytope, that admits similar constructions.
I will try to explain how the lineup polytope of the hypersimplex arises in quantum physics, related to a generalization of Pauli’s exclusion principle.
This is joint work with Arnau Padrol, Federico Castillo, Jean-Philippe Labbé and physicists Christian Schilling and Julia Liebert.