Chow Functions for Partially Ordered Sets

In a landmark paper in 1992, Stanley developed the foundations of what is now known as the Kazhdan--Lusztig--Stanley (KLS) theory. To each kernel in a graded poset, he associates special functions called KLS polynomials. This unifies and puts a common ground for i) the Kazhdan--Lusztig polynomial of a Bruhat interval in a Coxeter group, ii) the toric g-polynomial of a polytope, iii) the Kazhdan-Lusztig polynomial of a matroid. In this talk, I will introduce a new family of functions, called Chow functions, which encode various deep cohomological aspects of the combinatorial objects named before. In the three aforementioned settings, the Chow function describes i) a descent-like statistic enumerator for paths in the Bruhat graph, ii) the enumeration of chains of faces of the polytope, iii) the Hilbert series of the matroid Chow ring.

This is joint work with Jacob P. Matherne and Lorenzo Vecchi.