ISSN:
1572-9168
Keywords:
52B11
;
52B40
;
05B35
;
matroid
;
oriented matroid
;
reorientation class
;
d-cube
;
d-crosspolytope
;
Folkman-Lawrence representation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00181184
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