On the cube problem of Las Vergnas

• Type: Article in International Scientific Journal
• Date: 1996

Authors

Bokowski, J (Author) Other; deOliviera, AG (Author) FCUP; Thiemann, U (Author) Other; daCosta, AV (Author) Other

Journal

Title: Geometriae Dedicata

Vol. 63

Pages: 25-43

ISSN: 0046-5755

Publisher: Springer Nature

Indexing

ISI Web of Knowledge - 6 Citations

Scopus - 5 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-001-DE9

Abstract (EN): We prove a conjecture of Las Vergnas in dimensions d less than or equal to 7: The matroid of the d-dimensional cube C-d has a unique reorientation class. This extends a result of Las Vergnas, Roudneff and Salaun in dimension 4. Moreover, we determine the automorphism group G(d) of the matroid of the d-cube C-d for arbitrary dimension d, and we discuss its relation to the Coxeter group of C-d. We introduce matroid facets of the matroid of the d-cube in order to evaluate the order of G(d). These matroid facets turn out to be arbitrary pairs of parallel, subfacets of the cube. We show that the Euclidean automorphism group W-d is a proper subgroup of the group G(d) of all matroid symmetries of the d-cube by describing genuine matroid symmetries for each Euclidean facet. A main theorem asserts that any one of these matroid symmetries together with the Euclidean Coxeter symmetries generate the full automorphism group G(d). For the proof of Las Vergnas' conjecture we use essentially these symmetry results together with the fact that the reorientation class of an oriented matroid is determined by the labeled lower rank contractions of the oriented matroid. We also describe the Folkman-Lawrence representation of the vertex figure of the d-cube and a contraction of it. Finally, we apply our method of proof to show a result of Las Vergnas, Roudneff, and Salaun that the matroid of the 24-cell has a unique reorientation class, too.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 19