The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes

Abstract

Abstract Gelfand–Tsetlin polytopes arise in representation theory and algebraic combinatorics. One can construct the Gelfand–Tsetlin polytope

$$\mathrm{GT}_\lambda $$

GT λ

for any partition

$$\lambda = (\lambda _1,\ldots ,\lambda _n)$$

λ = (

λ 1

, … ,

λ n

)

of weakly increasing positive integers. The integral points in a Gelfand–Tsetlin polytope are in bijection with semi-standard Young tableau of shape

$$\lambda $$

λ

and parametrize a basis of the

$$\mathrm{GL}_n$$

GL n

-module with highest weight

$$\lambda $$

λ

. 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,

$$\mathrm{diam}(\mathrm{GT}_\lambda )$$

diam (

GT λ

)

, and the combinatorial automorphism group,

$$\mathrm{Aut}(\mathrm{GT}_\lambda )$$

Aut (

GT λ

)

, of any Gelfand–Tsetlin polytope. We exhibit two vertices that are separated by at least

$$\mathrm{diam}(\mathrm{GT}_\lambda )$$

diam (

GT λ

)

edges and provide an algorithm to construct a path of length at most

$$\mathrm{diam}(\mathrm{GT}_\lambda )$$

diam (

GT λ

)

between any two vertices. To identify the automorphism group, we study

$$\mathrm{GT}_\lambda $$

GT λ

using combinatorial objects called

$$ladder diagrams $$

ladderdiagrams

and examine faces of co-dimension 2.