We study the Hilbert–Poincaré series of three algebraic objects arising in the Chow-theoretic and Kazhdan–Lusztig framework of matroids. These are, respectively, the Hilbert–Poincaré series of the Chow ring, the augmented Chow ring, and the intersection cohomology module. We develop and highlight an explicit parallelism between the Kazhdan–Lusztig polynomial of a matroid and the Hilbert–Poincaré series of its Chow ring that extends naturally to the Hilbert–Poincaré series of both the intersection cohomology module and the augmented Chow ring.