GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

Abstract

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$M$.