We explore the Hodge theory behind the fact that the basis generating polynomial of a matroid is Lorentzian. The story reveals a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or vector configuration). After giving an overview of the similarity, I will outline proofs of two combinatorial conjectures, the nonnegativity conjecture for Kazhdan–Lusztig polynomials of matroids and the top-heavy conjecture for the number of flats of matroids. The key step is to formulate and prove an analogue of the decomposition theorem in a combinatorial setup. The talk will be accessible to graduate students. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.