Short rewriting, and geometric explanations related to the active bijection, for: Extension-lifting bijections for oriented matroids, by S. Backman, F. Santos, C.H. Yuen, arXiv:1904.03562v2 (October 29, 2023)

2023/12/12 by Emeric Gioan, Gioan, Emeric
Computer Science · Engineering · #Advanced Graph Theory Research #Advanced Optical Network Technologies #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2312.07522

openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an oriented matroid M, and given a generic single element extension and a generic single element lifting of M, the main result of [1] provides a bijection between bases of M and certain reorientations of M induced by the extension-lifting. This note is intended to somehow clarify and precise the geometric setting for this paper in terms of oriented matroid arrangements and oriented matroid programming, to describe and prove the main bijective result in a short simple way, and to show how it consists of combining two direct bijections and a central bijection, which is the same as a special case - practically uniform - of the bounded case of the active bijection [5, 6]. (The relation with the active bijection is addressed in [1] in an indirect and more complicated way.)

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• Extension-lifting Bijections for Oriented Matroids
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• The active bijection for graphs
• On Tutte polynomial expansion formulas in perspectives of matroids and\n oriented matroids