Combinatorics Seminar: "The polytope algebra of generalized permutahedra and the deformation cone of the type B permutahedron."

Abstract: Danilov-Koshevoy and, independently, Postnikov showed that any generalized permutahedron is a signed Minkowski sum of the faces of the standard simplex. In other words, they form a maximal linearly independent collection of rays in the deformation cone of the permutahedron. In contrast, Ardila-Castillo-Eur-Postnikov observed that the faces of the cross-polytope only span a subspace of roughly half the dimension of the deformation cone of the type B permutahedron. In this talk, we use McMullen’s polytope algebra to help explain this phenomenon and obtain a maximal collection of linearly independent rays for this cone. Concretely, we consider the subalgebra generated by deformations of a fixed zonotope Z and endow it with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We use this module structure to extract information about the deformation cone of Z and, in the case of the (type B) permutahedron, to relate it to certain Eulerian statistics on (signed) permutation.

Host: Laura Escobar Vega