Combinatorics & Geometry BLT Seminar

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.