Combinatorics of posets with a view towards geometry

Finite partially ordered sets (posets) abound throughout mathematics. Examples include the set of faces of a convex polytope ordered by inclusion, the bond lattice of a graph, the poset of flats of a matroid, and the Bruhat order on a Coxeter group. Often, to a relevant class of posets one can associate special polynomial invariants. For instance, the polynomial whose coefficient of degree i counts the number of chains of i elements in the poset arises naturally when studying barycentric subdivisions of polytopes.

In this course, we will explore a general class of polynomial invariants known as Kazhdan–Lusztig–Stanley (KLS) and Chow functions. The main motivation stems from the fact that these functions appear in several areas of mathematics with an algebro-geometric flavor—for example, they generalize the famous Kazhdan–Lusztig polynomials, the toric g-polynomials, and the Poincaré polynomials of Chow rings of matroids. Despite their deep connections, the theory of KLS and Chow functions can be developed with only a modest background in linear algebra and combinatorics, and without any prior knowledge of abstract algebra or algebraic geometry. We will focus on posets arising from familiar combinatorial structures such as polytopes and graphs, and we will learn what information the KLS and Chow functions encode.

We plan to follow material covered in the following sources:

1. Luis Ferroni, Jacob P. Matherne, and Lorenzo Vecchi, "Chow functions for partially ordered sets".
2. Luis Ferroni, Jacob P. Matherne, Matthew Stevens, and Lorenzo Vecchi, "Hilbert-Poincaré series of matroid Chow rings and intersection cohomology", Adv. Math. 449 (2024), Paper No. 109733, 55 pp.
3. Nicholas Proudfoot, "The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials", EMS Surv. Math. Sci. 5 (2018), no.1-2, 99–127.
4. Richard P. Stanley, "Subdivisions and local h-vectors" J. Amer. Math. Soc. 5 (1992), no.4, 805–851.