Vertices of Gelfand-Tsetlin Polytopes

Details Vertices of Gelfand-Tsetlin Polytopes Vertices of Gelfand-Tsetlin Polytopes

2003-09-19

arxiv.org/abs/math/0309329v2

Mathematics

Combinatorics

14 pages, 3 figures, fixed attributions

Scientific paper

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\mathfrak{gl}_n \C$ 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 non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.

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