The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes
Abstract Gelfand–Tsetlin polytopes arise in representation theory and algebraic combinatorics. One can construct the Gelfand–Tsetlin polytope $$\mathrm{GT}_\lambda $$...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131507 |
| _version_ | 1826203777572536320 |
|---|---|
| author | Gao, Yibo Krakoff, Benjamin Yang, Lisa |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gao, Yibo Krakoff, Benjamin Yang, Lisa |
| author_sort | Gao, Yibo |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:43:11Z |
| format | Article |
| id | mit-1721.1/131507 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:43:11Z |
| publishDate | 2021 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1315072023-11-08T21:34:46Z The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes Gao, Yibo Krakoff, Benjamin Yang, Lisa Massachusetts Institute of Technology. Department of Mathematics 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. 2021-09-20T17:17:22Z 2021-09-20T17:17:22Z 2019-04-09 2020-09-24T21:22:54Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131507 en https://doi.org/10.1007/s00454-019-00076-z Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media, LLC, part of Springer Nature application/pdf Springer US Springer US |
| spellingShingle | Gao, Yibo Krakoff, Benjamin Yang, Lisa The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title | The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title_full | The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title_fullStr | The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title_full_unstemmed | The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title_short | The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes |
| title_sort | diameter and automorphism group of gelfand tsetlin polytopes |
| url | https://hdl.handle.net/1721.1/131507 |
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