Keywords

**Abstract** Starting with a matroid $M$ one can construct three important algebraic objects: the Chow ring, the augmented Chow ring and the intersection cohomology. They play instrumental roles within the proofs of the log-concavity of the Whitney numbers of the first kind, and the top-heaviness of the Whitney numbers of the second kind. We will describe the combinatorics of their Hilbert series for matroids in general and uniform matroids in particular. Several aspects such as unimodality, gamma-positivity and real-rootedness will be discussed under different lenses. In particular, the Koszulness of the Chow ring and the augmented Chow ring seem to be a hint towards the real-rootedness conjectures.