Let
$M$
be a regular matroid. The Jacobian group
$\text{Jac}(M)$
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)
$\operatorname{Jac}(G)$
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
$\text{Jac}(G)$
and spanning trees. However, most of the known bijections use vertices of
$G$
in some essential way and are inherently ‘nonmatroidal’. 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
$\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$
between
${\mathcal{G}}(M)$
and bases of
$M$
. (Here
$\unicode[STIX]{x1D70E}$
(respectively
$\unicode[STIX]{x1D70E}^{\ast }$
) is an acyclic signature of the set of circuits (respectively cocircuits) of
$M$
.) We then give a geometric interpretation of each such map
$\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$
in terms of zonotopal subdivisions which is used to verify that
$\unicode[STIX]{x1D6FD}$
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
$M$
.
Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials
