# The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes

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## Abstract

Gelfand–Tsetlin polytopes arise in representation theory and algebraic combinatorics. One can construct the Gelfand–Tsetlin polytope \(\mathrm{GT}_\lambda \) for any partition \(\lambda = (\lambda _1,\ldots ,\lambda _n)\) of weakly increasing positive integers. The integral points in a Gelfand–Tsetlin polytope are in bijection with semi-standard Young tableau of shape \(\lambda \) and parametrize a basis of the \(\mathrm{GL}_n\)-module with highest weight \(\lambda \). The combinatorial geometry of Gelfand–Tsetlin polytopes has been of recent interest. Researchers have created new combinatorial models for the integral points and studied the enumeration of the vertices of these polytopes. In this paper, we determine the exact formulas for the diameter of the 1-skeleton, \(\mathrm{diam}(\mathrm{GT}_\lambda )\), and the combinatorial automorphism group, \(\mathrm{Aut}(\mathrm{GT}_\lambda )\), of any Gelfand–Tsetlin polytope. We exhibit two vertices that are separated by at least \(\mathrm{diam}(\mathrm{GT}_\lambda )\) edges and provide an algorithm to construct a path of length at most \(\mathrm{diam}(\mathrm{GT}_\lambda )\) between any two vertices. To identify the automorphism group, we study \(\mathrm{GT}_\lambda \) using combinatorial objects called \(ladder diagrams \) and examine faces of co-dimension 2.

## Keywords

Gelfand–Tsetlin Polytope Automorphism group Diameter## Mathematics Subject Classification

05E10 05E18## Notes

### Acknowledgements

This research was carried out as part of the 2016 REU program at the School of Mathematics at University of Minnesota, Twin Cities, and was supported by NSF RTG Grant DMS-1148634 and NSF Grant DMS-1351590. The authors are especially grateful to Victor Reiner for his mentorship and support, and for many fruitful conversations. The authors would also like to thank Elise delMas and Craig Corsi for their valuable advice and comments. Finally, we would like to thank the reviewers for their help in clarifying and improving the presentation of our arguments.

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