The radial derivative on the graded Möbius algebra

Abstract: Let $M$ be a simple matroid and let $B(M)$ be the graded Möbius 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 $D_β$ which acts as ordinary differentiation on a canonical ``radial'' copy of a truncated polynomial algebra. Allowing both $D_β$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series [ H_{β,M}(q)=\sum_{k=0}^r h_k^β(M)q^k. ] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_β$. Hence $H_β$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_β$ 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.

• The paper constructs the canonical lowering operator $D_\beta$ and the Hilbert series $H_\beta$, producing a lattice-invariant matroid profile that extends the classical apolar series while remaining bounded by Whitney numbers.
• The paper proves that $H_\beta$ is not valuative by separating matroids with identical Derksen $\mathcal G$-invariants, and its degree-two coefficients distinguish Bonin’s matroid pair.
• Exact computations for all 950 simple matroids on eight elements, plus modular tests on 400 nine-element matroids and theta-graph families, support log-concavity and front-loading conjectures that remain theoretically open.

The paper constructs a new numerical invariant of simple matroids from a differential calculus on the graded Möbius algebra $B(M)$, proves that this invariant is not valuative, and assembles substantial computational evidence for two shape conjectures. The construction is motivated by the recent failure of strong shape expectations for Whitney numbers of the second kind: Larson's graphic counterexamples to Mason's log-concavity conjecture and the non-unimodality results of Divoux–Larson–Lowen–Wang. Against that backdrop, the author proposes a canonical Hilbert series $H_{\beta,M}(q)$ that provably separates matroids indistinguishable by all valuative data and whose low-degree coefficients satisfy log-concavity exactly where the classical flat counts fail.

The radial core and the operator $D_\beta$

Let $L=L(M)$ be the lattice of flats of a rank-$r$ simple matroid, and let $B(L)$ be its graded Möbius algebra. For an atom $a$, denote by $L_a$ the multiplication operator $e_a\diamond(-)$, and let $\beta_x$ be the number of ordered atom bases of a flat $H_{\beta,M}(q)$0. Expanding $H_{\beta,M}(q)$1 gives

$H_{\beta,M}(q)$2

and the span of the $H_{\beta,M}(q)$3 is a graded subalgebra isomorphic to $H_{\beta,M}(q)$4 — the "radial core" of $H_{\beta,M}(q)$5. The paper equips $H_{\beta,M}(q)$6 with the weighted inner product in which the flat basis vectors are orthogonal with norms $H_{\beta,M}(q)$7, and lets $H_{\beta,M}(q)$8 be the adjoint of $H_{\beta,M}(q)$9. Under the injective map $D_\beta$0 sending $D_\beta$1 to the normalized basis-generating polynomial $D_\beta$2 of the flat, each $D_\beta$3 becomes ordinary partial differentiation $D_\beta$4 on the squarefree algebra; this realizes the adjoints as coordinate derivatives.

The central object is the degree-one lowering operator

$D_\beta$5

where $D_\beta$6, $D_\beta$7 multiplies by rank, and $D_\beta$8 multiplies by the number $D_\beta$9 of atoms not below $L=L(M)$0. It satisfies $L=L(M)$1, so it restricts to $L=L(M)$2 on the radial core. Two structural facts establish canonicity: away from the core, $L=L(M)$3 acts via the rescaled formula $L=L(M)$4 with $L=L(M)$5 — up to the factor $L=L(M)$6, the transpose of the Markov kernel of the canonical random extension process — and any diagonal normalization forcing $L=L(M)$7 must use precisely $L=L(M)$8. The construction is therefore choice-free and invariant under lattice isomorphism, though the author concedes that $L=L(M)$9 is neither claimed unique among extensions of $r$0 nor a derivation of the diamond product.

The cyclic module and its Hilbert series

Starting from $r$1, the module $r$2 is generated under $r$3 and the commuting family $r$4, graded by word length, with $r$5. Omitting $r$6 recovers the Macaulay inverse system of the basis-generating polynomial $r$7, whose Hilbert function $r$8 is symmetric (Artinian Gorenstein). The basic comparison

$r$9

positions $B(L)$0 as a canonical enlargement of a symmetric Gorenstein Hilbert function bounded by the top-heavy Whitney numbers. Notably, the reverse inequality $B(L)$1 fails for 938 of the 950 eight-element matroids, so $B(L)$2 is genuinely a different profile rather than a repackaging of flat counts.

Separation from valuative invariants

The strongest structural result concerns Bonin's pair of rank-three matroids with equal Derksen $B(L)$3-invariant but different configurations. Dualizing to simple rank-five matroids $B(L)$4, exact rational computation yields identical classical apolar series $B(L)$5 yet distinct series

$B(L)$6

Since every valuative invariant factors through $B(L)$7 (Derksen–Fink), $B(L)$8 cannot be the restriction to simple matroids of any valuative invariant; in particular it is not determined by the Tutte polynomial or catenary data. The separation mechanism is explicit and occurs already in degree two: modulo the common 25-dimensional space of pair-incidence vectors, the element-incidence vectors have ranks 5 for $B(L)$9 versus 3 for $a$0. A further consequence is that duality is not recoverable from $a$1: the original rank-three pair has $a$2 while their duals are separated. For uniform matroids, by contrast, $a$3 is redundant and $a$4.

The generalized theta family

The most pointed test comes from Larson's counterexample to Whitney log-concavity, the cycle matroid $a$5 of the theta graph with four internally disjoint paths of lengths $a$6; Larson's example is $a$7. The paper computes the first four coefficients for all $a$8:

$a$9	$L_a$0
0	$L_a$1
1	$L_a$2
2	$L_a$3
3	$L_a$4

and proves $L_a$5. For $L_a$6 the gap is $L_a$7. Thus the $L_a$8 inequality holds strictly at the differential index corresponding exactly to where Whitney log-concavity fails classically.

The proof combines representation theory of the symmetry group $L_a$9 with finite determinant witnesses. Classical degrees are handled via Gottlieb's incidence-matrix rank theorem applied to spanning-subset occupancy matrices, including a boundary case at $e_a\diamond(-)$0 resolved through the Specht decomposition of $e_a\diamond(-)$1. The nonclassical quotient $e_a\diamond(-)$2 is identified as $e_a\diamond(-)$3, where $e_a\diamond(-)$4 and $e_a\diamond(-)$5, with saturation established by compressed witness matrices whose determinants are explicit nonzero polynomials in $e_a\diamond(-)$6 (e.g., $e_a\diamond(-)$7 in the hardest sector). Words with two or three occurrences of $e_a\diamond(-)$8 are shown to add nothing, using the fact that $e_a\diamond(-)$9 is classical and $\beta_x$0 lies in the trivial isotypic component, which is already saturated.

Computational evidence

Exact rational computation over the complete Mayhew–Royle census verifies strict log-concavity and front-loading for all 950 simple matroids on eight elements, with $\beta_x$1 in 894 cases, smallest log-concavity gap 35, and largest observed defect $\beta_x$2 versus $\beta_x$3 — confirming the evidence is not driven by degenerate cases where $\beta_x$4 acts trivially. A stratified sample of 400 nine-element matroids was checked by a three-prime modular backend ($\beta_x$5), validated against the exact implementation on all 950 eight-element cases; the author is careful to state that three-prime agreement is evidence, not a rational-rank certificate, since modular reduction can only lower rank.

Limitations and open questions

The paper is candid that the computations do not constitute a structural theory: "at present we do not have a satisfying theoretical framework that explains the shape suggested by these computations." Both main conjectures — log-concavity of $\beta_x$6 and front-loading ($\beta_x$7 for $\beta_x$8, where $\beta_x$9) — remain open beyond the verified ranges. Three specific questions are posed: whether the associated graded of the $H_{\beta,M}(q)$00-filtration carries a Lorentzian or Lefschetz structure in the sense of recent work on Lefschetz modules; whether front-loaded defect can be realized geometrically by maps between complementary defect spaces; and how the filtered module behaves under direct sum, deletion, contraction, and duality, given that simple formulas fail and the Bonin pair shows duality is not determined by $H_{\beta,M}(q)$01 alone. One should also note that the theta-family analysis covers only degrees zero through three, so even within that family the full log-concavity chain is unverified.

Conclusion

This paper introduces a canonical lowering operator $H_{\beta,M}(q)$02 on the graded Möbius algebra, built from $H_{\beta,M}(q)$03-weighted adjoints and forced by a diagonal normalization, and uses it to define a graded cyclic module whose Hilbert series $H_{\beta,M}(q)$04 is a new, non-valuative matroid invariant. The invariant strictly refines the Derksen $H_{\beta,M}(q)$05-invariant and the classical apolar Hilbert function simultaneously, and it satisfies the critical log-concavity inequality at the precise index where Larson's counterexample breaks Whitney log-concavity. With complete verification on eight elements and strong modular evidence at nine, the outstanding problem is structural: finding the Hodge-theoretic or Lefschetz-type mechanism behind the observed shape.