[Paper Review] The Active Bijection 2.a - Decomposition of activities for matroid bases, and Tutte polynomial of a matroid in terms of beta invariants of minors

Emeric Gioan, Michel Las Vergnas|arXiv (Cornell University)|Jul 17, 2018

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper introduces the active filtration/partition of a matroid basis, a canonical decomposition of bases into minors with uniactive (1/0 or 0/1) internal or external activity. It derives a new expression for the Tutte polynomial in terms of beta invariants of minors, refining classical basis-activity and orientation-activity formulas through a structural decomposition based on the fundamental bipartite graph and active closure operator.

ABSTRACT

We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the active closure. It relies only on the fundamental bipartite graph of the basis and can be expressed also as a decomposition of general bipartite graphs on a linearly ordered set of vertices. From this, first, structurally, we obtain that the set of all bases can be canonically partitioned and decomposed in terms of such bases of minors induced by filtrations. Second, enumeratively, we derive an expression of the Tutte polynomial of a matroid in terms of beta invariants of minors. This expression refines at the same time the classical expressions in terms of basis activities and orientation activities (if the matroid is oriented), and the well-known convolution formula for the Tutte polynomial. Third, in a companion paper of the same series (No. 2.b), we use this decomposition of matroid bases, along with a similar decomposition of oriented matroids, and along with a bijection in the 1/0 activity case from a previous paper (No. 1), to define the canonical active bijection between orientations/signatures/reorientations and spanning trees/simplices/bases of a graph/real hyperplane arrangement/oriented matroid, as well as various related bijections.

Motivation & Objective

• To develop a canonical decomposition of matroid bases into minors with uniactive internal or external activity.
• To refine the classical Tutte polynomial expression in terms of basis activities by introducing a hierarchical decomposition via filtrations.
• To unify and generalize existing formulas for the Tutte polynomial, including basis-activity and orientation-activity expressions.
• To establish a structural foundation for the active bijection between orientations and bases in oriented matroids, as developed in companion papers.
• To define and characterize the active closure operator, which generates the active filtration from the fundamental bipartite graph of a basis.

Proposed method

• Introduces filtrations—nested sequences of subsets of the ground set—that induce minors via restriction and contraction.
• Defines the active closure operator on the ground set using the fundamental bipartite graph of a basis, enabling construction of the active filtration.
• Constructs the active filtration via a single-pass algorithm over the linearly ordered ground set, assigning each element to a component based on closure and activity conditions.
• Expresses the Tutte polynomial as a sum over all connected filtrations, with each term being a product of beta invariants of minors corresponding to internal and external uniactive bases.
• Uses the beta invariant β(M) as a count of uniactive internal or external bases in a minor, and β*(M) for the dual, to refine the Tutte polynomial coefficients.
• Establishes a canonical bijection between bases of the original matroid and tuples of uniactive bases in the minor sequence, grounded in the active filtration.

Experimental results

Research questions

• RQ1 How can the set of all bases of a matroid be canonically partitioned and decomposed in terms of minors with uniactive internal or external activity?
• RQ2 Can the Tutte polynomial be expressed as a sum over filtrations, with each term involving beta invariants of minors?
• RQ3 How does the active closure operator relate to the fundamental bipartite graph of a basis and enable a unique decomposition of bases?
• RQ4 In what way does this decomposition refine both the classical basis-activity formula and the orientation-activity formula for the Tutte polynomial?
• RQ5 What is the structural and enumerative significance of the active filtration in the context of oriented matroids and the active bijection?

Key findings

• The active filtration of a basis is a unique, canonical decomposition into a sequence of minors, each with a basis that is either internally or externally uniactive (activity 1/0 or 0/1).
• The Tutte polynomial of a matroid is expressed as a sum over all connected filtrations, with each term being a product of beta invariants of minors: $ t(M;x,y) = extstyleigsum igl( extstyleigprod_{k=1}^{ heta} eta(M(F_k)/F_{k-1}) igr) igl( extstyleigprod_{k=1}^{ ho} eta^{*}(M(F'_k)/F'_{k-1}) igr) x^{ heta} y^{ ho} $.
• The coefficient of $ x^{ heta} y^{ ho} $ in the Tutte polynomial equals the number of bases with internal activity $ heta $ and external activity $ ho $, which is in bijection with tuples of uniactive bases in the minor sequence induced by a connected filtration.
• The beta invariant $ eta(M) $ counts the number of uniactive internal bases in $ M $, and $ eta^{*}(M) $ counts the number of uniactive external bases in $ M $, generalizing Zaslavsky’s and Las Vergnas’s results on bounded regions.
• The active closure operator, defined via the fundamental bipartite graph, allows a constructive, single-pass algorithm to compute the active filtration, ensuring consistency and uniqueness.
• The decomposition refines both the classical basis-activity formula and the orientation-activity formula for oriented matroids, providing a unifying algebraic and structural framework.