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Monotone Paths on Zonotopes and Oriented Matroids

Published online by Cambridge University Press:  20 November 2018

Christos A. Athanasiadis  and
Francisco Santos
Christos A. Athanasiadis
Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden. email: athana@math.kth.se
Francisco Santos
Affiliation:
Departamento de Matemàticas, Estadística y Computación, Universidad de Cantabria, Santander, E-39005, Spain. email: santos@matesco.unican.es
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Abstract

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Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\left\lceil 2n/\left( n-d+2 \right) \right\rceil -1$ flips for all $n\ge d+2\ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d\ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.

Keywords

52C3552B1252C4020F55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 6 , 01 December 2001 , pp. 1121 - 1140
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Athanasiadis, C. A., Edelman, P. H. and Reiner, V., Monotone paths on polytopes. Math. Z. 235(2000), 315–334.Google Scholar
[2] Barnette, D., Graph theorems for manifolds. Israel J. Math. 16(1973), 62–72.Google Scholar
[3] Billera, L. J. and Sturmfels, B., Fiber polytopes. Ann. of Math. 135(1992), 527–549.Google Scholar
[4] Billera, L. J., Kapranov, M. M. and Sturmfels, B., Cellular strings on polytopes. Proc. Amer. Math. Soc. 122(1994), 549–555.Google Scholar
[5] Björner, A., Vergnas, M. Las, Sturmfels, B., White, N. and Ziegler, G. M., Oriented Matroids. Encyclopedia Math. Appl. 46, Cambridge University Press, Cambridge, 1993; second edition, 1999.Google Scholar
[6] Bourbaki, N., Groupes et algébres de Lie. Ch. IV–VI, Hermann, Paris, 1968.Google Scholar
[7] Cordovil, R. and Fukuda, K., Oriented matroids and combinatorial manifolds. European J. Combin. 14(1993), 9–15.Google Scholar
[8] Cordovil, R. and Moreira, M. L., A homotopy theorem on oriented matroids. Discrete Math. 111(1993), 131–136.Google Scholar
[9] Deligne, P., Les immeubles des groupes de tresses généralisés. Invent. Math. 17(1972), 273–302.Google Scholar
[10] Edelman, P. H., A partial order on the regions of ℝ n dissected by hyperplanes. Trans. Amer. Math. Soc. 283(1984), 617–631.Google Scholar
[11] Edelman, P. H. and Reiner, V., personal communication, 1999.Google Scholar
[12] Humphreys, J. E., Reflection groups and Coxeter groups. Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, England, 1990.Google Scholar
[13] Orlik, P. and Terao, H., Arrangements of Hyperplanes. Grundlehren Math. Wiss. 300, Springer-Verlag, New York, 1992.Google Scholar
[14] Reiner, V., The generalized Baues problem. In New Perspectives in Algebraic Combinatorics (eds. Billera, L. J. et al.), MSRI Book Series 38, Cambridge University Press, New York, 1999, 293–336.Google Scholar
[15] Santos, F., A point set whose space of triangulations is disconnected. J. Amer. Math. Soc. 13(2000), 611–637.Google Scholar
[16] Ziegler, G. M., Lectures on Polytopes. Graduate Texts in Math. 152, Springer-Verlag, New York, 1995.Google Scholar