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.
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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
An, B.H., Cho, Y., Kim, J.S.: On the f-vectors of Gelfand–Cetlin polytopes. ArXiv e-prints. (2016). arXiv:1606.05957
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
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)
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
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
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
Geissinger, L.: The face structure of a poset polytope. In: Proceedings of the Third Caribbean conference in combinatorics and computing, pp. 125–133 (1981)
Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71, 825–828 (1950)
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
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
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
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
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
Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986). https://doi.org/10.1007/BF02187680
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
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Pegel, C. The Face Structure and Geometry of Marked Order Polyhedra. Order 35, 467–488 (2018). https://doi.org/10.1007/s11083-017-9443-2
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DOI: https://doi.org/10.1007/s11083-017-9443-2