Title: Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory,
Abstract: Let $M$ be a regular matroid. The Jacobian group ${\rm Jac}(M)$ 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) ${\rm Jac}(G)$ 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 ${\rm Jac}(G)$ 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-to-describe 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 ${\mathcal
G}(M)$ of circuit-cocircuit reversal classes of $M$, and then define a family
of combinatorial bijections $\beta_{\sigma,\sigma^*}$ between ${\mathcal G}(M)$
and bases of $M$. (Here $\sigma$ (resp. $\sigma^*$) is an acyclic signature of
the set of circuits (resp. cocircuits) of $M$.) We then give a geometric
interpretation of each such map $\beta=\beta_{\sigma,\sigma^*}$ in terms of
zonotopal subdivisions which is used to verify that $\beta$ 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 $M$.