  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. We develop a parallelism between the
Kazhdan-Lusztig polynomial of a matroid and the Hilbert series of its Chow
ring. This extends to a parallelism between the $Z$-polynomial of a matroid and
the Hilbert 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;
also, by leveraging that they are valuations under matroid polytope
subdivisions, we deduce a fast way for computing them for a large class of
matroids. Uniform matroids are a case of combinatorial interest; we link the
resulting polynomials with certain real-rooted families such as 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 result of Haglund and Zhang; and
in addition, we prove a version of a conjecture of Gedeon in the Chow setting:
uniform matroids maximize coefficient-wisely these polynomials for matroids
with fixed rank and size. By relying on the nonnegativity 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
$\gamma$-positivity, a property for palindromic polynomials that lies between
unimodality and real-rootedness; this settles a conjecture of Ferroni, Nasr,
and Vecchi.
