Geometric objects for regular orthogonal matroid BBY bijections

Develop geometric objects corresponding to the zonotope constructions used for regular matroids in the more general setting of regular orthogonal matroids, with the aim of providing a geometric proof of the bijectivity of the BBY map.

Background

The paper generalizes BBY torsors and BBY bijections from regular matroids to regular orthogonal matroids. In the regular-matroid case, bijectivity can be proved using geometric reasoning involving tilings of a zonotope. The authors state that they do not know how to define analogous geometric objects for regular orthogonal matroids. They therefore use a purely combinatorial approach developed in earlier work, so the geometric formulation remains unresolved in the paper.

References

It is not clear how to define the corresponding geometric objects in the more general setting of regular orthogonal matroids, so we employ the machinery introduced in the second author's paper , in which a purely combinatorial proof of the bijectivity of $\bar{\beta}$ is given.

— The Jacobian of a regular orthogonal matroid and torsor structures on spanning quasi-trees of ribbon graphs  (2501.08796 - Baker et al., 15 Jan 2025) in Section 1, subsection “A few words on the proof of Theorem 2”; repeated in Section 5, opening paragraph