[Paper Review] Valuative invariants for large classes of matroids

Luis Ferroni, Benjamin Schröter|arXiv (Cornell University)|Aug 9, 2022

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper introduces a novel framework based on relaxing stressed subsets to systematically study valuative invariants across large classes of matroids, particularly split matroids. It establishes that evaluations of key invariants—such as the Tutte polynomial, Ehrhart polynomial, Kazhdan–Lusztig polynomials, and Chow rings—on split matroids depend entirely on their behavior on a tractable subclass called cuspidal matroids, via a unified matroid subdivision of the hypersimplex.

ABSTRACT

We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a \emph{stressed subset}. This framework provides a new combinatorial characterization of the class of split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which in turn can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on split matroids depend solely on the behavior of the invariant on tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan--Lusztig polynomials, the Whitney numbers of the first and second kind, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's $g$-polynomials, as well as Chow rings of matroids and their Hilbert--Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

Motivation & Objective

• To develop a general method for computing valuative invariants on large classes of matroids, especially split matroids.
• To provide a combinatorial characterization of split matroids through the operation of relaxing stressed subsets.
• To unify the computation of diverse invariants—such as Ehrhart polynomials, Tutte polynomials, and Kazhdan–Lusztig polynomials—by reducing them to evaluations on a tractable subclass: cuspidal matroids.
• To demonstrate that the structure of the hypersimplex subdivision encodes the behavior of valuative invariants across matroid polytopes.
• To offer a computational tool for generating explicit data and concrete examples of invariants across matroid classes.

Proposed method

• Introduce the operation of relaxing a stressed subset to transform a matroid into another with more bases, enabling controlled matroid subdivisions.
• Construct a matroid subdivision of the hypersimplex where each cell corresponds to a matroid, with base polytopes of split matroids appearing as cells.
• Leverage the valuative property of invariants to reduce evaluations on split matroids to evaluations on a subclass of Schubert matroids known as cuspidal matroids.
• Use the Chow ring of the universal matroid $\mathsf{U}_{n,n}$ as a universal target for valuative invariants, via a canonical map $\Theta$ from the matroid polytope group.
• Apply the $\mathcal{G}$-invariant and its factorization through the Chow ring to reconstruct invariants from catenary data and flag structures.
• Utilize the fact that any valuative invariant vanishing on matroids with loops factors through the Chow ring map $\Theta$, enabling a universal description.

Experimental results

Research questions

• RQ1 How can valuative invariants on split matroids be computed in a unified and systematic way?
• RQ2 What is the role of stressed subsets and their relaxations in constructing matroid subdivisions of the hypersimplex?
• RQ3 To what extent do evaluations of invariants like the Tutte polynomial or Ehrhart polynomial on split matroids depend only on a smaller, tractable subclass of matroids?
• RQ4 Can the Chow ring of the universal matroid $\mathsf{U}_{n,n}$ serve as a universal receptacle for all valuative invariants?
• RQ5 How does the structure of the hypersimplex subdivision reflect the behavior of invariants such as Kazhdan–Lusztig polynomials and $g$-polynomials?

Key findings

• Evaluations of any valuative invariant on split matroids are completely determined by their values on the subclass of cuspidal matroids.
• The hypersimplex admits a matroid subdivision in which the base polytope of any split matroid appears as a cell, enabling explicit computation of invariants.
• The map $\Theta: \mathfrak{M}_E \to \underline{\mathrm{CH}}(\mathsf{U}_{n,n})$ provides a universal factorization for all valuative invariants vanishing on matroids with loops.
• The $\mathcal{G}$-invariant of a loopless matroid factors through the Chow ring of $\mathsf{U}_{n,n}$, allowing reconstruction of invariants from flag data and catenary vectors.
• The framework unifies and explains recent results on invariants such as the Ehrhart polynomial, Tutte polynomial, and Kazhdan–Lusztig polynomials across matroid classes.
• The approach provides a powerful computational tool for generating explicit data and concrete examples of invariants, particularly for split and cuspidal matroids.