Let M be a regular matroid. The Jacobian group Jac(M ) of M is a finite abelian group whose cardinality is equal to the number of bases of M . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) Jac(G) of a graph G (in which case bases of the corresponding regular matroid are spanning trees of G).There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph Jac(G) and spanning trees. However, most of the known bijections use vertices of G in some essential way and are inherently "non-matroidal". In this paper, we construct a family of explicit and easy-todescribe bijections between the Jacobian group of a regular matroid M and bases of M , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of M admits a canonical simply transitive action on the set G(M ) of circuit-cocircuit reversal classes of M , and then define a family of combinatorial bijections β σ,σ * between G(M ) and bases of M . (Here σ (resp. σ * ) is an acyclic signature of the set of circuits (resp. cocircuits) of M .) We then give a geometric interpretation of each such map β = β σ,σ * in terms of zonotopal subdivisions which is used to verify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zonotope Z; by passing to dilations we obtain a new derivation of Stanley's formula linking the Ehrhart polynomial of Z to the Tutte polynomial of M . arXiv:1701.01051v3 [math.CO] 23 Apr 20191 If e and e * are dual edges of G and G * , respectively, then given an orientation for e * we orient e by rotating the orientation of e * clockwise locally near the crossing of e and e * .

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The Jacobian group Jac(G) of a finite graph G is a group whose cardinality is the number of spanning trees of G. G also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et al. obtained a polyhedral decomposition of the tropical Jacobian where vertices and cells correspond to elements of Jac(G) and spanning trees of G, respectively. We give a combinatorial description of bijections coming from this geometric setting. This provides a new geometric method for constructing bijections in combinatorics. We introduce a special class of geometric bijections that we call edge ordering maps, which have good algorithmic properties. Finally, we study the connection between our geometric bijections and the class of bijections introduced by Bernardi; in particular we prove a conjecture of Baker that planar Bernardi bijections are "geometric". We also give sharpened versions of results by Baker and Wang on Bernardi torsors. arXiv:1509.02963v2 [math.CO]