Vertices of Gelfand-Tsetlin polytopes

J.A. De Loera, T.B. McAllister

    Research output: Contribution to journalArticleAcademicpeer-review

    23 Citations (Scopus)

    Abstract

    This paper is a study of the polyhedral geometry of Gelfand–Tsetlin polytopes arising in the representation theory of ${\frak gl}_n \Bbb C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand–Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand–Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when $n$ is fixed.
    Original languageEnglish
    Pages (from-to)459-470
    JournalDiscrete and Computational Geometry
    Volume32
    Issue number4
    DOIs
    Publication statusPublished - 2004

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