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Coarsening polyhedral complexes

Author: Nathan Reading
Journal: Proc. Amer. Math. Soc. 140 (2012), 3593-3605
MSC (2010): Primary 52B99, 52C35
Published electronically: February 23, 2012
MathSciNet review: 2929028
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Abstract: Given a pure, full-dimensional, locally strongly connected polyhedral complex  $ \mathcal {C}$, we characterize, by a local codimension-$ 2$ condition, polyhedral complexes that coarsen  $ \mathcal {C}$. The proof of the characterization draws upon a general shortcut for showing that a collection of polyhedra is a polyhedral complex and upon a property of hyperplane arrangements which is equivalent, for Coxeter arrangements, to Tits' solution to the Word Problem. The motivating special case, the case where  $ \mathcal {C}$ is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietze's convexity theorem.

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  • 1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler, Oriented matroids (Second edition), Encyclopedia of Mathematics and its Applications, 46, Cambridge Univ. Press, 1999. MR 1744046 (2000j:52016)
  • 2. R. Cordovil and M. L. Moreira, A homotopy theorem on oriented matroids. Discrete Math. 111 (1993), no. 1-3, 131-136. MR 1210090 (94d:52016)
  • 3. P. Deligne, Les immeubles des groupes de tresses généralisés. Invent. Math. 17 (1972), 273-302. MR 0422673 (54:10659)
  • 4. J. Morton, L. Pachter, A. Shiu, B. Sturmfels and O. Wienand, Convex rank tests and semigraphoids. SIAM J. Discrete Math. 23 (2009), no. 3, 1117-1134. MR 2538642 (2011b:62126)
  • 5. N. Reading, Lattice congruences, fans and Hopf algebras. J. Combin. Theory Ser. A 110 (2005), no. 2, 237-273. MR 2142177 (2006b:20054)
  • 6. M. Salvetti, Topology of the complement of real hyperplanes in $ C^N$. Invent. Math. 88 (1987), no. 3, 603-618. MR 884802 (88k:32038)
  • 7. H. Tietze, Bemerkungen über konvexe und nicht-konvexe Figuren. J. Reine Angew. Math. 160 (1929), 67-69.
  • 8. J. Tits, Le problème des mots dans les groupes de Coxeter. 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, 175-185. Academic Press, London. MR 0254129 (40:7339)
  • 9. F. A. Valentine, Convex sets. McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264 (30:503)
  • 10. G. Ziegler, Lectures on polytopes. Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995. MR 1311028 (96a:52011)

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Additional Information

Nathan Reading
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

Received by editor(s): May 6, 2010
Received by editor(s) in revised form: April 14, 2011
Published electronically: February 23, 2012
Additional Notes: The author was partially supported by NSA grant H98230-09-1-0056.
Communicated by: Jim Haglund
Article copyright: © Copyright 2012 Nathan Reading

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