Vertices of Gelfand--Tsetlin Polytopes (English)

In: Discrete & Computational Geometry   ;  32 ,  4  ;  459-470  ;  2004

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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.

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Guest Editors#8217 Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
Guest Editors’ Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
The Partial-Fractions Method for Counting Solutions to Integral Linear Systems
Beck, Matthias | 2004
Isoradial Bodies
Brandenberg, René / Dattasharma, Abhi / Gritzmann, Peter / Larman, David | 2004
Vertices of Gelfand--Tsetlin Polytopes
Loera, Jesús A. / McAllister, Tyrrell B. | 2004
A Complexity Bound on Faces of the Hull Complex
Develin, Mike | 2004
Enumerative Properties of Ferrers Graphs
Ehrenborg, Richard / Willigenburg, Stephanie | 2004
Tchebyshev Posets
Hetyei, Gábor | 2004
Mixed Fibre Polytopes
McMullen, Peter | 2004
Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices
Nyman, Kathryn / Swartz, Ed | 2004
Convex Combinatorial Optimization
Onn, Shmuel / Rothblum, Uriel G. | 2004
Non-Crossing Frameworks with Non-Crossing Reciprocals
Orden, David / Rote, Günter / Santos, Francisco / Servatius, Brigitte / Servatius, Herman / Whiteley, Walter | 2004
The Et-Construction for Lattices, Spheres and Polytopes
Paffenholz, Andreas / Ziegler, Günter M. | 2004
Graphs, Syzygies, and Multivariate Splines
Rose, Lauren L. | 2004