Séminaire Lotharingien de Combinatoire, 82B.39 (2019), 12 pp.

Spencer Backman, Francisco Santos, and Chi Ho Yuen

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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

The following versions are available: