Discrete Geometry and Combinatorics Seminar
Abstract: We introduce the Primitive Eulerian polynomial PA(z) of a central hyperplane arrangement A, a reparametrization of its cocharacteristic polynomial. Previous work on the polytope algebra of deformations of a zonotope (2021) implicitly showed that this polynomial has nonnegative coefficients whenever A is a simplicial arrangement, but a combinatorial interpretation of the coefficients was only found for reflection arrangements of type A and B. We discuss the relationship between the Primitive Eulerian polynomial and the usual Eulerian polynomial. We also present a geometric/combinatorial interpretation for the coefficients of PA(z) for all simplicial arrangements A, along with some real-rootedness results and conjectures. Based on joint work with Christophe Hohlweg and Franco Saliola.