Chow Rings of Matroids and Atomistic Lattices

After Feichner and Yuzvinsky introduced the Chow ring associated to ranked atomistic lattices in 2003, little study of them was made before Adiprisito, Huh, and Katz used them to resolve the long-standing Heron-Rota-Walsh conjecture, proving along the way that the Chow rings of geometric lattices satisfy versions of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Here, we seek to remedy the lack of basic knowledge about the Chow rings of atomic lattices by providing some general techniques for computing their Hilbert series, by making detailed study a few fundamental examples, and by providing a number of interesting conjectures based on our observations. Using the incidence algebra, we give a compact formula for the Hilbert series of Chow rings associated to both ranked atomic lattices and products of them. In a special case, we define a generalization of the Hilbert series and give a formula for the Hilbert series of the product in terms of differential operators. In addition to general techniques, we study in detail the Chow rings associated to the lattices of flats of uniform and linear matroids. We show that the Hilbert series of uniform and linear matroids take forms of combinatorial interest; in particular, the Hilbert series of the linear matroid associated to an n-dimensional vector space over a finite field Fq can be described in terms of the q-Eulerian polynomial defined by Shareshian and Wachs in [SW10], and the Hilbert series of the Chow ring of a uniform matroid can be described in terms of elementary statistics on Sn. We also compute the Charney-Davis quantities of the rings, which come out to linear combinations of the secant numbers in the uniform case, and to a linear combination of the q-secant numbers of Foata and Han in the case of the linear matroids. Finally, we assert that Poincaré duality holds for the Chow rings a a slightly more general class of ranked atomistic lattices than those studied by Adiprasito, Huh, and Katz, and make a conjecture about the class of ranked atomic lattices for which Poincaré duality holds.