Hilbert–Poincaré series of matroid Chow rings and intersection cohomology

We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan–Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. (The last two are known as the Z-polynomial and the Kazhdan–Lusztig polynomial, respectively.) We develop an explicit parallelism between the Kazhdan–Lusztig polynomial of a matroid and the Hilbert–Poincaré series of its Chow ring. This extends to a parallelism between the Z-polynomial of a matroid and the Hilbert–Poincaré series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Our two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them. We provide several intrinsic definitions of these invariants via recursions they satisfy. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families appearing in combinatorics such as the Eulerian and the binomial Eulerian polynomials, and we settle a conjecture of Hameister, Rao, and Simpson. Furthermore, we prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids via a technique introduced by Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wise these polynomials for matroids with fixed rank and cardinality. By relying on the nonnegativity of the coefficients of the Kazhdan–Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, we strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to γ-positivity, a property for palindromic polynomials that lies between unimodality and real-rootedness; this also settles a conjecture of Ferroni, Nasr, and Vecchi.