The Circuit-Cocircuit Reversal System and Torsor Structures on Spanning Trees

The Jacobian of a finite graph G is a finite abelian group whose cardinality is equal to the number of spanning trees of G.  In this earlier post, I discussed a family of combinatorial bijections between elements of {\rm Jac}(G) and the set {\mathcal T}(G) of spanning trees of G.  I also discussed the fact that for plane graphs, these Bernardi bijections come from a natural simply transitive action of {\rm Jac}(G) on {\mathcal T}(G) (or, more precisely, a natural isomorphism class of such actions).

In the present post, I’d like to discuss a different family of simply transitive actions of {\rm Jac}(G) on {\mathcal T}(G) discovered by myself, Spencer Backman (a former student of mine), and Chi Ho Yuen (a current student of mine).  One virtue of our construction is that it generalizes in a natural way from graphs to regular matroids.  (We will mostly stick to the graphical case in this post, but will insert some comments about extensions to regular and/or oriented matroids.  A regular matroid can be thought of, rather imprecisely, as the smallest natural class of objects which contain graphs and admit a duality theory generalizing duality for planar graphs. Regular matroids are special cases of the more general concept of oriented matroids.)

One of the main new ideas in [BBY] (as we will henceforth refer to our paper) is to use the torsor {\rm Pic}^{g-1}(G) as an intermediate object rather than {\rm Pic}^{g}(G).  The latter is canonically isomorphic (as a {\rm Jac}(G)-torsor) to the set of break divisors on G; the former is isomorphic to the circuit-cocircuit reversal system, which we now introduce.

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