The intersection cohomology module $\mathrm{IH}(\mathcal{L};\mathbb{Q})$ of a matroid $\mathcal{L}$ is a module over the graded Möbius algebra $\mathrm{H}(\mathcal{L};\mathbb{Q})$. For a matroid arising from a hyperplane arrangement, it is isomorphic to the intersection cohomology of an "arrangement Schubert variety", as a module over its cohomology ring. This module played a central role in the proofs of the Dowling-Wilson top-heaviness conjecture and of the nonnegativity of coefficients of matroid Kazhdan-Lusztig polynomials. I will discuss the key properties of this module, and how it is constructed. I will also explain a generalization to coefficients other than $\mathbb{Q}$, called a "parity module" because it corresponds to sections of a parity sheaf on the arrangement Schubert variety.

Joint work with June Huh, Jacob Matherne, Nick Proudfoot and Botong Wang.