Résumé
The active bijection for oriented matroids (and real hyperplane arrangements,
and graphs, as particular cases) is introduced and investigated by the authors
in a series of papers. Given any oriented matroid defined on a linearly ordered
ground set, we exhibit one particularité of its bases, which we call its
active basis, with remarkable properties. It preserves activities (for oriented
matroids in the sense of Las Vergnas, for matroid bases in the sense of Tutte),
as well as some active partitions of the ground set associated with oriented
matroids and matroid bases. It yields a canonical bijection between classes of
reorientations and bases [...]. It also yields a refined bijection between all
reorientations and subsets of the ground set. Those bijections are related to
various Tutte polynomial expressions [...]. They contain various noticeable
bijections involving orientations/signatures/reorientations and spanning
trees/simplices/bases of a graph/real hyperplane arrangement/oriented matroid.
[...]
In previous papers of this series, we defined the active bijection between
bounded regions and uniactive internal bases by means of fully optimal bases
(No. 1), and we defined a decomposition of activities for matroid bases by
means of [...] particular sequences of minors (companion paper, No. 2.a). The
present paper is central in the series. First, we define a decomposition of
activities for oriented matroids, using the same sequences of minors, yielding
a decomposition of an oriented matroid into bounded regions of minors. Second,
we use the previous results together to provide the canonical and refined
active bijections alluded to above. We also give an overview and examples of
the various results of independent interest involved in the construction. They
arise as soon as the ground set of an oriented matroid is linearly ordered.