Speaker:   Derek Hanely, University of Kentucky

Title:          Ehrhart Theory of Paving and Panhandle Matroids



Abstract:



There has been a wealth of research recently on polytopes arising from matroids. One such polytope, the matroid base polytope, is obtained as the convex hull of incidence vectors corresponding to the bases of an underlying matroid. In this talk, we will discuss a generalization of Ferroni's work on the Ehrhart theory of sparse paving matroids. In particular, we will show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of \textit{stressed-hyperplane relaxation} introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. To conclude, we will present evidence that panhandle matroids are Ehrhart positive and, as an application of the main result, compute the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes. This is joint work with Jeremy Martin, Daniel McGinnis, Dane Miyata, George Nasr, Andrés Vindas Meléndez, and Mei Yin.