Combinatorics & Geometry BLT Seminar

Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank r on n elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants.  In this talk, we will first sketch the background that is needed to understand their polytope and valuative invariants.  We will then provide the needed characterizations of many of the matroids that Ferroni and Fink conjectured to yield vertices, we will give additional examples of such matroids, and we will mention some open problems.

Order polytopes of posets have been a very rich topic at the crossroads between combinatorics and discrete geometry since their definition by Stanley in 1986. The h^*-polynomials of order polytopes of graded posets are known to be gamma-nonnegative by work of Brändén, but it is currently unknown whether they are real-rooted (this is a special case of the Neggers-Stanley conjecture, that was disproved in its general form by work of Brändén and Stembridge).

In the present talk we will introduce an equivariant version of Brändén's gamma-nonnegativity result, using the tools of equivariant Ehrhart theory (introduced by Stapledon in 2011). Namely, we prove that order polytopes of graded posets are always gamma-effective, i.e., that the gamma-polynomial associated with the equivariant h^∗-polynomial of the order polytope of any graded poset has coefficients consisting of actual characters. This is joint work with Akihiro Higashitani.