### 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 language | English |
---|---|

Pages (from-to) | 459-470 |

Journal | Discrete and Computational Geometry |

Volume | 32 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2004 |

## Fingerprint Dive into the research topics of 'Vertices of Gelfand-Tsetlin polytopes'. Together they form a unique fingerprint.

## Cite this

De Loera, J. A., & McAllister, T. B. (2004). Vertices of Gelfand-Tsetlin polytopes.

*Discrete and Computational Geometry*,*32*(4), 459-470. https://doi.org/10.1007/s00454-004-1133-3