Work

Characterisations of strong $Δ$-matroids

Year: 2026

Type: preprint

Abstract: We study characterisations of strong $Δ$-matroids, compiling a list of five equivalent descriptions. We show a variant of Wenzel's exchange property and the hyperplane exchange property of Borovik-Gelfand-White are equivalent. We also introduce two novel characterisations in terms of 'peerless' and 'isolated' antipodes within the system of feasible sets, banning certain configurations of antipodes either globally or locally. As a corollary, we obtain new 'local' exchange axioms for matroids and $Δ$-matroids. We give algebraic motivation for these new characterisations by introducing the peerless antipode equations, tropical equations that govern whether a $Δ$-matroid has no peerless antipodes. We show that these arise as the tropicalisation of a specific basis of quadratics cutting out the orthogonal Grassmannian.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2607.02419

Authors: Calvert, Kieran, Dermenjian, Aram, Fink, Alex, Smith, Ben

Institutions: Queen Mary University of London, Lancaster University, Universidad de Sevilla

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Engineering and Physical Sciences Research Council, Agencia Estatal de Investigación

Awards: G1175115578, Extensions of matroid Hodge theory, G2262748287, G2281333282, G2390397592, Abstract rigidity for natural stability problems, G300979063, G34346518, G3480869486, G4084979677, G425668148, G4526231891, G5080475149, G586857820, G6200043914, G7084143925

OpenAlex ID: W7167253531

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