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.
Computing linear extensions and order polynomials of posets is in general a hard problem with no explicit formulas and nice structure and properties. When the poset is a Young diagram of a straight shape, the number of linear extensions is given by the hook-lnegth formula, yet the corresponding number of plane partitions (order polynomial of that poset) does not have a product formula. In the absence of such nice formulas we will show a general approach to proving that the order polynomials of posets corresponding to skew Young diagrams have positive coefficients. We will also discuss some applications to matroids. Joint work with Luis Ferroni and Alejandro Morales.
Zaslavsky’s theorem says that the number of regions in the complement of a real hyperplane arrangement is equal to an evaluation of the characteristic polynomial of the arrangement. This relation generalises to all oriented matroids and can be categorified using the Orlik-Solomon algebra and three seemingly different filtrations of the tope space of an oriented matroid: the dual Varchenko–Gelfand degree filtration, Kalinin’s spectral sequence, and Quillen’s augmentation filtration. In this talk, we show that all of these filtrations coincide. The Varchenko–Gelfand filtration has the advantage that it is defined over the integers. We also show that the Varchenko-Gelfand approach can be used to filter the so-called Z-sign cosheaf on fan of the underlying matroid. The filtration of the Z/2-sign cosheaf has previous applications to the topology of real algebraic varieties via patchworking.
This talk is based on joint work with Chi Ho Yuen.