The radial derivative on the graded M\"obius algebra

Abstract

Let $M$ be a simple matroid and let $B(M)$ be the graded M\"obius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical global lowering operator $D_\beta$ which acts as ordinary differentiation on a canonical ``radial'' copy of a truncated polynomial algebra. Allowing both $D_\beta$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series \[ H_{\beta,M}(q)=\sum_{k=0}^r h_k^\beta(M)q^k. \] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_\beta$. Hence $H_\beta$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_\beta$ is log-concave and top-heavy in differential degree. For the generalized theta family containing Larson's counterexample to Whitney log-concavity, we compute the first four coefficients and prove the critical log-concavity inequality. Exact computation verifies both conjectures for all $950$ simple matroids on eight elements.