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, Volume 35, Issue 3, pp 467–488 | Cite as

The Face Structure and Geometry of Marked Order Polyhedra

  • Christoph Pegel
Article
  • 36 Downloads

Abstract

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand–Tsetlin polytopes and cones, as well as Berenstein–Zelevinsky polytopes—all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.

Keywords

Marked poset polytopes Gelfand–Tsetlin polytopes Polyhedral geometry Representation theory 

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References

  1. 1.
    Alexandersson, P.: Gelfand–Tsetlin polytopes and the integer decomposition property. Eur. J. Comb. 54, 1–20 (2016).  https://doi.org/10.1016/j.ejc.2015.11.006 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, B.H., Cho, Y., Kim, J.S.: On the f-vectors of Gelfand–Cetlin polytopes. ArXiv e-prints. (2016). arXiv:1606.05957
  3. 3.
    Ardila, F., Bliem, T., Salazar, D.: Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as marked poset polytopes. J. Combinatorial Theor. Ser. A 118(8), 2454–2462 (2011).  https://doi.org/10.1016/j.jcta.2011.06.004 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cahill, J., Fickus, M., Mixon, D.G., Poteet, M.J., Strawn, N.K.: Constructing finite frames of a given spectrum and set of lengths. Appl. Comput. Harmon. Anal. 35(1), 52–73 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    De Loera, A.J., Mcallister, B.T.: Vertices of Gelfand–Tsetlin polytopes. Discrete Comput. Geom. 32(4), 459–470 (2004).  https://doi.org/10.1007/s00454-004-1133-3 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fang, X., Fourier, G.: Marked chain-order polytopes. Eur. J. Comb. 58, 267–282 (2016).  https://doi.org/10.1016/j.ejc.2016.06.007 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fourier, G.: Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence. J. Pure and Appl. Algebra 220(2), 606–620 (2016).  https://doi.org/10.1016/j.jpaa.2015.07.007 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Geissinger, L.: The face structure of a poset polytope. In: Proceedings of the Third Caribbean conference in combinatorics and computing, pp. 125–133 (1981)Google Scholar
  9. 9.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71, 825–828 (1950)MathSciNetGoogle Scholar
  10. 10.
    Gusev, P., Kiritchenko, V., Timorin, V.: Counting vertices in Gelfand–Zetlin polytopes. J. Combinatorial Theor. Ser. A 120(4), 960–969 (2013).  https://doi.org/10.1016/j.jcta.2013.02.003 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haga, T., Pegel, C.: Polytopes of eigensteps of finite equal norm tight frames. Discrete Comput. Geom. 56(3), 727–742 (2016).  https://doi.org/10.1007/s00454-016-9799-x MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hibi, T., Li, N.: Unimodular equivalence of order and chain polytopes. Mathematica Scandinavica 118(1), 5–12 (2016).  https://doi.org/10.7146/math.scand.a-23291 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hibi, T., Li, N., Sahara, Y., Shikama, A.: The numbers of edges of the order polytope and the chain poyltope of a finite partially ordered set. ArXiv e-prints. (2015). arXiv:1508.00187
  14. 14.
    Jochemko, K., Sanyal, R.: Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings. SIAM J. Discret. Math. 28(3), 1540–1558 (2014).  https://doi.org/10.1137/130944849 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986).  https://doi.org/10.1007/BF02187680 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Stanley, R.P., Pitman, J.: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Comput. Geom. 27(4), 603–602 (2002).  https://doi.org/10.1007/s00454-002-2776-6 MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut für Algebra, Zahlentheorie und Diskrete MathematikLeibniz Universität HannoverHannoverGermany

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