The Hodge theory of matroids by Adiprasito--Huh--Katz resolved outstanding conjectures in matroid theory by establishing positivity properties for Chow rings of matroids. In algebraic geometry, a notion complementary to the Chow ring is the K-ring of a variety. We establish positivity properties for K-rings of matroids. Our results give a new proof of the nonnegativity of the omega invariant of a matroid, in support of Speyer's f-vector conjecture, and resolve the conjecture of Tohǎneanu that higher order Orlik--Terao algebras are Cohen--Macaulay.
