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Coxeter-associahedra

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Victor Reiner  and
Günter M. Ziegler
Victor Reiner
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Günter M. Ziegler
Affiliation:
Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Heilbronner Str. 10, D-10711 Berlin, Germany
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Abstract

Recently M. M. Kapranov [Kap] defined a poset KPAn−1, called the permuto-associahedron, which is a hybrid between the face poset of the permutohedron and the associahedron. Its faces are the partially parenthesized, ordered, partitions of the set {1, 2, …, n}, with a natural partial order.

Kapranov showed that KPAn−1, is the face poset of a regular CW-ball, and explored its connection with a category-theoretic result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equations, and a certain moduli space of curves. He also asked the question of whether this CW-ball can be realized as a convex polytope.

We show that indeed, the permuto-associahedron corresponds to the type An−1, in a family of convex polytopes KPW associated to the classical Coxeter groups, W = An−1, Bn, Dn. The embedding of these polytopes relies on the secondary polytope construction of the associahedron due to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral coordinates, with all vertices on a sphere, and include a complete description of the facet-defining inequalities.

Also we show that for each W, the dual polytope KPW* is a refinement (as a CW-complex) of the Coxeter complex associated to W, and a coarsening of the barycentric subdivision of the Coxeter complex. In the case W = An−1, this gives a combinatorial proof of Kapranov's original sphericity result.

MSC classification

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 364 - 393
Copyright
Copyright © University College London 1994

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References

BS1.Billera, L. J. and Sturmfels, B.. Fiber polytopes. Annals of Mathematics, 135 (1992), 527–549.Google Scholar
BS2.Billera, L. J. and Sturmfels, B.. Iterated fiber polytopes. Mathematika, 41 (1994), 348–363.CrossRefGoogle Scholar
Bj1.Björner, A.. Shellable and Cohen-Macaulay partially ordered sets. Transactions Amer. Math. Soc, 260 (1980), 159–183.CrossRefGoogle Scholar
Bj2.Björner, A.. Posets, regular CW complexes and Bruhat order. European J. Combinatorics, 5 (1984), 7–16.CrossRefGoogle Scholar
BLSWZ.Björner, A., Las Vergnas, M., Sturmfels, B., White, N. and Ziegler, G. M.. Oriented Matroids. Encyclopedia of Mathematics (Cambridge University Press, 1993).Google Scholar
BW.Björner, A. and Wachs, M.. On lexicographically shellable posets. Transactions Amer. Math. Soc, 277 (1983), 323–341.CrossRefGoogle Scholar
Bro.Brown, K. S.. Buildings (Springer-Verlag, New York, 1989).CrossRefGoogle Scholar
Chr.Christof, Th.. Ein Verfahren zur Transformation zwischen Polyederdarstellungen. Diplomarbeit, Universitat Augsburg, 1991, 79 pages, plus manual and listing.Google Scholar
CJR.Christof, Th., Jünger, M. and Reinelt, G.. A complete description of the traveling salesman polytope on 8 nodes. Operations Research Letters, 10 (1990), 497–500.Google Scholar
DK.Danaraj, G. and Klee, V.. Shellings of spheres and polytopes. Duke Math. J., 41 (1974), 443–451.CrossRefGoogle Scholar
GZK1.Gel'fand, I. M., Zelevinsky, A. V. and Kapranov, M. M.. Newton polytopes of principal A-determinants, Soviet Math. Doklady, 40 (1990), 278–281.Google Scholar
GZK2.Gel'fand, I. M., Zelevinsky, A. V. and Kapranov, M. M.. Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Leningrad Math. Journal, 2 (1991), 449–505.Google Scholar
GrZ.Gritzmann, P. and Sturmfels, B.. Minkowski addition of polytopes: Computational complexity and applications to Gröbner bases, SI AM J. Discrete Math., 6 (1993), 246–269.CrossRefGoogle Scholar
GrP.Grötschel, M. and Padberg, M.. Polyhedral Theory. Chapter 8 in: Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., Schmoys, D. B. (eds.), The Traveling Salesman Problem (Wiley, 1985), 251–360.Google Scholar
Hai.Haiman, M.. Constructing the associahedron. Unpublished manuscript, (MIT, 1984), 11 pages.Google Scholar
Hum.Humphreys, J. E.. Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, 29 (Cambridge University Press, 1990).CrossRefGoogle Scholar
Kap.Kapranov, M. M.. Permuto-associahedron, MacLane coherence theorem and asymptotic zones for the KZ equation. Journal of Pure and Applied Algebra, 85 (1993), 119–142.Google Scholar
Lee.Lee, C. W.. The associahedron and triangulations of the n-gon. European J. Combinatorics, 10 (1989), 551–560.Google Scholar
Mun.Munkres, J. R.. Elements of Algebraic Topology, (Addison-Wesley, Menlo Park, 1984).Google Scholar
Per.Perles, M. A.. List of problems. Conference on Convexity, Oberwolfach, July 1984.Google Scholar
Rei.Reiner, V.. Signed posets. J. Combinatorial Theory, Ser. A, 62 (1993), 324–360.Google Scholar
Sch.Schrijver, A.. Theory of Linear and Integer Programming, Wiley-Interscience Series in Discrete Mathematics and Optimization (John Wiley & Sons, Chichester, 1986).Google Scholar
Stan.Stanley, R. P.. Enumerative Combinatorics, Vol. I (Wadsworth & Brooks/Cole, Monterey, 1986).CrossRefGoogle Scholar
Stas.Stasheff, J. D.. Homotopy associativity of H-spaces I. Transactions Amer. Math. Soc, 108 (1963), 275–292.Google Scholar
Stu.Sturmfels, B.. Fiber polytopes: a brief overview. In: “Special Differential Equations” (ed. Yoshida, M.) (Kyushu University, Fukuoka, 1991), pp. 117–124.Google Scholar
Zie.Ziegler, G. M.. Lectures on Polytopes. Graduate Texts in Math., 152 (Springer, 1994).Google Scholar