A quadratic form generalization of rational dinv

Abstract: We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality [ Q(\mathbf{n})\geq \dfrac{1}{|G|}\,|\mathbf{n}|_\infty^2] if $\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.

• The paper demonstrates that the quadratic form Q recovers the Gorsky–Mazin dinv statistic for any rational Dyck path.
• It establishes positive definiteness of Q within a natural cone, ensuring convergence for related q-series in singularity theory.
• The work bridges combinatorial invariants with algebraic geometry and formal proof verification, opening avenues for high-rank generalizations.

Quadratic Form Generalization of Rational $\mathtt{dinv}$

Introduction and Motivation

The paper "A quadratic form generalization of rational dinv" (2604.13238) presents a new algebraic quadratic form $Q$ on the space of functions defined over the gap poset $G$ of the numerical semigroup $\langle a, b \rangle$, and establishes its combinatorial and geometric significance in rational Dyck path theory, particularly in connection with the Gorsky–Mazin $\mathtt{dinv}$ statistic and its higher-rank generalizations. The work demonstrates that $Q$, despite a definition involving negative coefficients, is effectively positive definite within a naturally arising cone $C_{\mathbb{R}}$ of decreasing real-valued functions on $G$. This positive definiteness plays a pivotal role in the convergence properties of certain $q$-series related to counting finite modules over plane curve singularities, and in suggesting Rogers–Ramanujan type identities in the context of moduli spaces on singular algebraic curves.

Definitions and Main Results

The Quadratic Form and Its Connection to $\mathtt{dinv}$

Let $Q$0 be the set of positive-integer lattice points $Q$1 satisfying $Q$2, with $Q$3 coprime positive integers, $Q$4. This realizes $Q$5 as the set of boxes beneath the diagonal in the $Q$6 rectangle: the combinatorial environment for rational Dyck paths.

The paper defines a quadratic form

$Q$7

where $Q$8, and a cone

$Q$9

For an indicator function $G$0 of a subdiagram (i.e., an upward-closed subset or rational Dyck path) $G$1, it is proven that $G$2 precisely recovers the Gorsky–Mazin $G$3 statistic. This result provides an algebraic characterization of $G$4 and solidifies a bridge between classical rational Dyck path enumeration and a more general quadratic framework.

Bilinear Form and Cross-$G$5

A symmetric bilinear form $G$6 associated with $G$7 is introduced. On pairs of subdiagrams $G$8, $G$9 computes a new quantity, the cross-$\langle a, b \rangle$0, a nonnegative combinatorial statistic, which generalizes $\langle a, b \rangle$1 to pairs of paths.

Strong results include:

• For any subdiagram $\langle a, b \rangle$2, $\langle a, b \rangle$3, and $\langle a, b \rangle$4 for nonempty $\langle a, b \rangle$5.
• For any subdiagrams $\langle a, b \rangle$6, $\langle a, b \rangle$7, which is symmetric and nonnegative.

Positivity and Quantitative Bounds

The core technical result is the demonstration of effective positive definiteness:

$\langle a, b \rangle$8

for all $\langle a, b \rangle$9. This ensures $\mathtt{dinv}$0 is nonnegative throughout the cone and strictly positive unless $\mathtt{dinv}$1 vanishes. The proof uses the decomposition of elements in $\mathtt{dinv}$2 as convex combinations of subdiagram indicator functions along with the positive combinatorics of cross-$\mathtt{dinv}$3.

Combinatorics and Algebraic Geometry Implications

The quadratic form $\mathtt{dinv}$4 is motivated by geometric models in the study of Quot schemes on singular plane curves $\mathtt{dinv}$5, arising in high-rank generalizations of the rational Dyck path story and their associated $\mathtt{dinv}$6-series. The positivity of $\mathtt{dinv}$7 enables convergence and finiteness properties for infinite $\mathtt{dinv}$8-series counting groupoid volumes of finite modules over the local rings $\mathtt{dinv}$9, weighted by automorphism group sizes. This result is essential for interpreting such sums as generating series for moduli problems in representation theory and singularity theory.

Furthermore, the expansion of $Q$0 to high-rank combinatorics and nested Dyck paths provides a concrete candidate for a "high-rank $Q$1," opening new connections to generalizations of the $Q$2-Catalan theory and likely to have ramifications for Khovanov–Rozansky–HOMFLY-PT link homology and Hilbert schemes.

The demonstration that $Q$3 is positive definite, with effective lower bounds, also directly supports conjectured sum=product Rogers–Ramanujan type identities for generating functions associated with these moduli, underpinning future research on explicit product and series expressions in algebraic combinatorics.

Formalization and Automated Theorem Proving

A notable aspect is the formal ground-truth verification of the central theorem via the Lean theorem prover and AxiomProver/AXLE pipeline, demonstrating a fully automated translation of the main results and their proofs into the formal language of mathematics. This emphasizes both the technical robustness of the results and the increasing role of machine-assisted theorem proving in advanced combinatorics and algebraic geometry.

Future Directions

The quadratic form framework suggests a systematic approach to generalizing rational Dyck path statistics to higher ranks and more general poset environments, potentially leading to a theory of nested path invariants and their geometric interpretations. The effective bounds raise questions about extremal configurations, potential spectral analysis of the quadratic form $Q$4, and applications in modular representation theory and $Q$5-series.

The paper closes with open questions regarding further combinatorial interpretations of cross-$Q$6, its connections to existing notions in the literature on high-rank and nested Dyck statistics, and the precise nature of the sum=product identities forecasted by this framework.

Conclusion

This work establishes a rigorous quadratic-form generalization of the rational $Q$7 statistic, recovers and extends key combinatorial invariants of Dyck path enumeration, and tightly links combinatorial algebraic structures with geometric and modular counting results in singularity theory. The analytic and formal verifications presented reinforce the foundational character of the results, with clear implications for future developments in algebraic combinatorics, geometric representation theory, and the formalization of higher rank symmetry statistics.