Abstract
We prove a conjecture of Las Vergnas in dimensions d≤7: The matroid of the d-dimensional cube C d has a unique reorientation class. This extends a result of Las Vergnas, Roudneff and Salaün 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 Salaün that the matroid of the 24-cell has a unique reorientation class, too.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Astie-Vidal, A.: The automorphism group of a matroid, in: Combinatorics 79 (Proceeding of a colloquium, Université de Montreal, Montreal, Quebec, 1979), Part II, Ann. Discrete Math. 9 (1980), 205–216.
Björner, A., Las Vergnas, M., Sturmfels, B., White, N. and Ziegler, G. M.: Oriented Matroids, Cambridge University Press, Cambridge, 1993.
Bokowski, J.: On recent progress in computational synthetic geometry, in POLYTOPES: Abstract, Convex and Computational, T. Bisztriczky, P. McMullen, R. Schneider and A. Ivić Weiss, Kluwer Academic Publishers, Series C: Mathematical and Physical Sciences 440 (1994), 335–358.
Bokowski, J.: Oriented matroids, in P. Gruber and J. M. Wills (eds), Handbook of Convex Geometry, Elsevier, North-Holland, Netherlands, 1993, 555–602.
Bokowski, J. and Guedes de Oliviera, A.: Invariant theory-like theorems for matroids and oriented matroids, Advances in Math. 109 (1994), 34–44.
Bokowski, J. and Sturmfels, B.: Computational Synthetic Geometry, Lecture Notes in Math. 1355, Springer, Heidelberg, 1989.
Cordovil, R., Guedes de Oliviera, A. and Fukuda, K.: On the cocircuit-graph of an oriented matroid, Preprint, 1991.
Coxeter, H. S. M. and Moser, W. O. J.: Generators and relations for discrete groups, Springer, Berlin, 1. Ausgabe, Ergeb. Math. Grenzgeb., Reihe: Gruppentheorie (Neue Folge Band 14), 1957.
Coxeter, H. S. M.: Regular and semi-regular polytopes I, Math. Z. 46 (1940), 380–407.
Coxeter, H. S. M. and Todd, J. A.: Abstract definitions for the symmetry groups of the regular polytopes in terms of two generators, Part I: The complete groups, Proc. Cambridge Philos. Soc. 32 (1936), 194–200.
Folkman, J. and Lawrence, J.: Oriented matroids, J. Combin. Theory, Ser. B, 25 (1978), 199–236.
Grove, L. C. and Benson, C. T.: Finite Reflection Groups, Springer, Heidelberg, 1985.
Grünbaum B.: Convex Polytopes, Interscience Publ., London, 1967.
Grünbaum B.: Arrangements and Spreads, Regional Conf., Vol. 10, 1972 (Amer. Math. Soc., Providence, RI, 1972).
Hudelson, M., Klee, V. and Larman, D.: Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem, Manuscript, 1994, 57pp.
Las Vergnas, M.: Matroödes orientables, preprint announced in C. R. Acad. Sci. Paris, Ser. A, 280 (1974), 61–64, laser synthetized published as: Bland, R. G. and Las Vergnas, M.: Orientability of matroids, J. comb. th. (B) 24 (1978), 94–123.
Las Vergnas, M., Roudneff, J.-P. and Salaün, I.: Regular polytopes and oriented matroids, Preprint, 1991.
Neumann, B. H.: Die Automorphismengruppe der freien Gruppen, Math. Ann. 107 (1933), 367–386.
Roudneff, J.-P.: Reconstruction of the orientation class of an oriented matroid. European J. Combin. 9 (1988), 423–429.
Salaün, I.: Deux problèmes de gèometrie combinatoire, Thèse de troisième Cycle, Université Paris 6, 1988.
Schönert, M. et al.: GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 4th edition, 1994.
White, N.: Theory of Matroids, Cambridge University Press, Cambridge, 1986.
Thiemann, U.: Kombinatorische Geometrien und Reorientierungsklassen, Diplom thesis, University Darmstadt, 1995.
Ziegler, G. M.: Some minimal non-orientable matroids of rank 3, Geom. Dedicata 38 (1991), 365–371.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bokowski, J., de Oliviera, A.G., Thiemann, U. et al. On the cube problem of Las Vergnas. Geom Dedicata 63, 25–43 (1996). https://doi.org/10.1007/BF00181184
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00181184