Topological bijections for oriented matroids

Abstract

In previous work by the first and third author with Matthew Baker, a family of bijections between bases of a regular matroid and the Jacobian group of the matroid was given. The core of the work is a geometric construction using zonotopal tilings that produces bijections between the bases of a realizable oriented matroid and the set of (σ, σ∗ )-compatible orientations with respect to some acyclic circuit (respectively, cocircuit) signature σ (respectively, σ∗). In this work, we extend this construction to general oriented matroids and circuit (respectively, cocircuit) signatures coming from generic single-element liftings (respectively, extensions). As a corollary, when both signatures are induced by the same lexicographic data, we give a new (bijective) proof of the interpretation of TM(1, 1) using orientation activity due to Gioan and Las Vergnas. Here TM(x, y) is the Tutte polynomial of the matroid.