2003/09/19 by Jesús A. De Loera, De Loera, Jesús A., Tyrrell B. McAllister +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #math.CO #math.RT
paper · pdf · doi:10.48550/arxiv.math/0309329
14 pages, 3 figures, fixed attributions
arxiv created 2003/09/23 · arxiv updated 2009/12/01
This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory \mathfrakgln \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≥5 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.