Skip to main contentSkip to search
Episciences
Open Access Journals
Sign in(new window)
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Sign in(new window)
Articles & Issues
All articlesAll accepted articlesAll volumesLast volumeSectionsSpecial issuesAuthors
About
The journalIndexing
Boards
Publish
For authorsEthical charter
Submit
Discrete Mathematics & Theoretical Computer Science logo
Contact
|
Credits
eISSN 1365-8050
|
RSS
|
Atom
Episciences
Documentation
|
Acknowledgements
|
Publishing policy
Accessibility: non-compliant
|
Legal mentions
|
Privacy statement
|
Terms of use
  1. Home > Articles & Issues >
  2. Articles >
  3. Gelfand―Tsetlin Poly ...
Conference paper

Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

Federico Ardila (1), Thomas Bliem (1), Dido Salazar (1)
(1) Department of Mathematics [San Francisco]
Download article
Open on HAL
Imported on
January 31, 2017
Published on
December 31, 2010
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
Proceedings
DOI
10.46298/dmtcs.2888
License
https://about.hal.science/hal-authorisation-v1
Creating Momentum through Communicating Mathematics
Indicators
733
Views
716
Downloads

Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

Federico Ardila (1), Thomas Bliem (1), Dido Salazar (1)
(1) Department of Mathematics [San Francisco]
Abstract
English
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
French
Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • poset
  • polytope
  • semisimple Lie algebra
  • PBW filtration
Cited by

Source: OpenCitations

  • DEGENERATE SCHUBERT VARIETIES IN TYPE A

    arXiv (Cornell University)

    Authors : ROCCO CHIRIVÌ ORCID, XIN FANG ORCID, GHISLAIN FOURIER ORCID

    Journal reference : Volume 26, 2020, pp. 1189-1215

    DOI : 10.1007/s00031-020-09558-4

  • Restricted Birkhoff Polytopes and Ehrhart Period Collapse

    Discrete & Computational Geometry

    Authors : Per Alexandersson ORCID, Sam Hopkins, Gjergji Zaimi

    Journal reference : Volume 73, 2023, pp. 62-78

    DOI : 10.1007/s00454-023-00611-z

  • A Continuous Family of Marked Poset Polytopes

    SIAM Journal on Discrete Mathematics

    Authors : Xin Fang ORCID, Ghislain Fourier ORCID, Jan-Philipp Litza, Christoph Pegel ORCID

    Journal reference : Volume 34, 2020, pp. 611-639

    DOI : 10.1137/18m1228529

  • Counting integer points in polytopes associated with directed graphs

    Advances in Applied Mathematics

    Authors : Ilse Fischer ORCID

    Journal reference : Volume 79, 2016, pp. 125-153

    DOI : 10.1016/j.aam.2016.04.008

  • A diagrammatic approach to string polytopes

    Algebraic Combinatorics

    Authors : Christian Steinert ORCID

    Journal reference : Volume 5, 2022, pp. 63-91

    DOI : 10.5802/alco.196

  • Newton–Okounkov Bodies of Flag Varieties and Combinatorial Mutations

    International Mathematics Research Notices

    Authors : Naoki Fujita ORCID, Akihiro Higashitani ORCID

    Journal reference : Volume 2021, 2020, pp. 9567-9607

    DOI : 10.1093/imrn/rnaa276

  • The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes

    arXiv (Cornell University)

    Authors : Yibo Gao, Benjamin Krakoff, Lisa Yang ORCID

    Journal reference : Volume 62, 2019, pp. 209-238

    DOI : 10.1007/s00454-019-00076-z

  • Vertices of FFLV polytopes

    Journal of Algebraic Combinatorics

    Authors : Evgeny Feigin ORCID, Igor Makhlin ORCID

    Journal reference : Volume 45, 2016, pp. 1083-1110

    DOI : 10.1007/s10801-016-0735-1

  • GELFAND–TSETLIN DEGENERATIONS OF REPRESENTATIONS AND FLAG VARIETIES

    arXiv (Cornell University)

    Authors : I. MAKHLIN ORCID

    Journal reference : Volume 27, 2020, pp. 563-596

    DOI : 10.1007/s00031-020-09622-z

  • Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence

    arXiv (Cornell University)

    Authors : Ghislain Fourier ORCID

    Journal reference : Volume 220, 2015, pp. 606-620

    DOI : 10.1016/j.jpaa.2015.07.007

  • Newton–Okounkov Polytopes of Flag Varieties for Classical Groups

    arXiv (Cornell University)

    Authors : Valentina Kiritchenko ORCID

    Journal reference : Volume 5, 2019, pp. 355-371

    DOI : 10.1007/s40598-019-00125-8

  • Polytopes and Large Counterexamples

    arXiv (Cornell University)

    Authors : Per Alexandersson ORCID

    Journal reference : Volume 28, 2017, pp. 115-120

    DOI : 10.1080/10586458.2017.1361367

  • $f$-vector inequalities for order and chain polytopes

    MATHEMATICA SCANDINAVICA

    Authors : Ragnar Freij-Hollanti ORCID, Teemu Lundström

    Journal reference : Volume 130, 2024, pp.

    DOI : 10.7146/math.scand.a-143995

  • Root, flow and order polytopes with connections to toric geometry

    Forum of Mathematics Sigma

    Authors : Konstanze Rietsch ORCID, Lauren Kiyomi Williams ORCID

    Journal reference : Volume 13, 2025, pp.

    DOI : 10.1017/fms.2025.10056

  • PBW-degenerated Demazure modules and Schubert varieties for triangular elements

    Journal of Combinatorial Theory Series A

    Authors : Ghislain Fourier ORCID

    Journal reference : Volume 139, 2015, pp. 132-152

    DOI : 10.1016/j.jcta.2015.12.001

  • Degrees of the stretched Kostka quasi‐polynomials

    Bulletin of the London Mathematical Society

    Authors : Shiliang Gao ORCID, Yibo Gao ORCID

    Journal reference : Volume 56, 2024, pp. 1748-1761

    DOI : 10.1112/blms.13024

  • Chain-order polytopes: toric degenerations, Young tableaux and monomial bases

    Algebraic Combinatorics

    Authors : Igor Makhlin ORCID

    Journal reference : Volume 7, 2024, pp. 1525-1550

    DOI : 10.5802/alco.384

  • FFLV-type monomial bases for type B

    Algebraic Combinatorics

    Authors : Igor Makhlin ORCID

    Journal reference : Volume 2, 2019, pp. 305-322

    DOI : 10.5802/alco.41

  • Two Double Poset Polytopes

    arXiv (Cornell University)

    Authors : Thomas Chappell, Tobias Friedl, Raman Sanyal ORCID

    Journal reference : Volume 31, 2017, pp. 2378-2413

    DOI : 10.1137/16m1091800

  • Relative poset polytopes and semitoric degenerations

    Selecta Mathematica

    Authors : Evgeny Feigin ORCID, Igor Makhlin ORCID

    Journal reference : Volume 30, 2024, pp.

    DOI : 10.1007/s00029-024-00935-5

  • PBW Filtration and Bases for Symplectic Lie Algebras

    arXiv (Cornell University)

    Authors : Evgeny Feigin ORCID, Ghislain Fourier ORCID, Peter Littelmann ORCID

    Journal reference : Volume 2011, 2011, pp. 5760-5784

    DOI : 10.1093/imrn/rnr014

  • PBW-type filtration on quantum groups of type A

    Journal of Algebra

    Authors : Xin Fang ORCID, Ghislain Fourier ORCID, Markus Reineke ORCID

    Journal reference : Volume 449, 2015, pp. 321-345

    DOI : 10.1016/j.jalgebra.2015.09.054

  • Restricted chain-order polytopes via combinatorial mutations

    European Journal of Combinatorics

    Authors : Oliver Clarke ORCID, Akihiro Higashitani, Francesca Zaffalon ORCID

    Journal reference : Volume 133, 2025, pp. 104326

    DOI : 10.1016/j.ejc.2025.104326

  • Birational maps, PBW degenerate flags and poset polytopes

    Journal of Algebra

    Authors : Evgeny Feigin ORCID

    Journal reference : Volume 674, 2025, pp. 235-256

    DOI : 10.1016/j.jalgebra.2025.02.039

  • Gelfand--Tsetlin Polytopes: A Story of Flow and Order Polytopes

    arXiv (Cornell University)

    Authors : Ricky I. Liu ORCID, Karola Mészáros, Avery St. Dizier ORCID

    Journal reference : Volume 33, 2019, pp. 2394-2415

    DOI : 10.1137/19m1251242

  • On Piecewise-Linear Homeomorphisms Between Distributive and Anti-blocking Polyhedra

    arXiv (Cornell University)

    Authors : Christoph Pegel ORCID, Raman Sanyal ORCID

    Journal reference : Volume , 2020, pp. 95-114

    DOI : 10.1007/978-3-030-52111-0_8

  • PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples

    arXiv (Cornell University)

    Authors : Igor Makhlin ORCID

    Journal reference : Volume 23, 2019, pp. 2315-2330

    DOI : 10.1007/s10468-019-09943-y

  • LINEAR RECURRENCE RELATIONS IN Q-SYSTEMS VIA LATTICE POINTS IN POLYHEDRA

    arXiv (Cornell University)

    Authors : CHUL-HEE LEE ORCID

    Journal reference : Volume 24, 2018, pp. 429-466

    DOI : 10.1007/s00031-018-9502-9

  • Linear Recursions for Integer Point Transforms

    Springer proceedings in mathematics & statistics

    Authors : Katharina Jochemko ORCID

    Journal reference : Volume , 2022, pp. 221-231

    DOI : 10.1007/978-3-030-98327-7_10

  • Beyond the Sottile–Sturmfels Degeneration of a Semi-Infinite Grassmannian

    arXiv (Cornell University)

    Authors : Evgeny Feigin ORCID, Igor Makhlin ORCID, Alexander Popkovich

    Journal reference : Volume 2023, 2022, pp. 10037-10066

    DOI : 10.1093/imrn/rnac116

  • Symmetries on plabic graphs and associated polytopes

    Comptes Rendus Mathématique

    Authors : Xin Fang ORCID, Ghislain Fourier ORCID

    Journal reference : Volume 356, 2018, pp. 581-585

    DOI : 10.1016/j.crma.2018.05.003

  • Gelfand–Tsetlin polytopes and the integer decomposition property

    European Journal of Combinatorics

    Authors : Per Alexandersson ORCID

    Journal reference : Volume 54, 2015, pp. 1-20

    DOI : 10.1016/j.ejc.2015.11.006

  • Tropical symplectic flag varieties: a Lie theoretic approach

    Algebraic Combinatorics

    Authors : George Balla ORCID, Xin Fang ORCID

    Journal reference : Volume 7, 2024, pp. 773-799

    DOI : 10.5802/alco.350

  • Kirillov–Reshetikhin crystals, energy function and the combinatorial R-matrix

    Journal of Algebraic Combinatorics

    Authors : Deniz Kus ORCID

    Journal reference : Volume 43, 2015, pp. 45-74

    DOI : 10.1007/s10801-015-0625-y

  • Marked chain-order polytopes

    European Journal of Combinatorics

    Authors : Xin Fang ORCID, Ghislain Fourier ORCID

    Journal reference : Volume 58, 2016, pp. 267-282

    DOI : 10.1016/j.ejc.2016.06.007

  • Combinatorial mutations of Gelfand–Tsetlin polytopes, Feigin–Fourier–Littelmann–Vinberg polytopes, and block diagonal matching field polytopes

    Journal of Pure and Applied Algebra

    Authors : Oliver Clarke ORCID, Akihiro Higashitani ORCID, Fatemeh Mohammadi ORCID

    Journal reference : Volume 228, 2024, pp. 107637

    DOI : 10.1016/j.jpaa.2024.107637

  • Coupled cluster degree of the Grassmannian

    Journal of Symbolic Computation

    Authors : Viktoriia Borovik ORCID, Bernd Sturmfels ORCID, Svala Sverrisdóttir ORCID

    Journal reference : Volume 128, 2024, pp. 102396

    DOI : 10.1016/j.jsc.2024.102396

  • Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces

    arXiv (Cornell University)

    Authors : Rekha Biswal, Ghislain Fourier

    Journal reference : Volume 18, 2015, pp. 1481-1503

    DOI : 10.1007/s10468-015-9548-5

  • The Face Structure and Geometry of Marked Order Polyhedra

    arXiv (Cornell University)

    Authors : Christoph Pegel ORCID

    Journal reference : Volume 35, 2017, pp. 467-488

    DOI : 10.1007/s11083-017-9443-2

  • On the combinatorics of string polytopes

    Journal of Combinatorial Theory Series A

    Authors : Yunhyung Cho ORCID, Yoosik Kim ORCID, Eunjeong Lee ORCID, Kyeong-Dong Park ORCID

    Journal reference : Volume 184, 2021, pp. 105508

    DOI : 10.1016/j.jcta.2021.105508

  • Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

    arXiv (Cornell University)

    Authors : Igor Makhlin ORCID

    Journal reference : Volume 185, 2021, pp. 105541

    DOI : 10.1016/j.jcta.2021.105541

  • Birational Maps to Grassmannians, Representations and Poset Polytopes

    Algebras and Representation Theory

    Authors : Evgeny Feigin ORCID

    Journal reference : Volume 27, 2024, pp. 1981-1999

    DOI : 10.1007/s10468-024-10273-x

  • Arithmetic of Marked Order Polytopes, Monotone Triangle Reciprocity, and Partial Colorings

    arXiv (Cornell University)

    Authors : Katharina Jochemko ORCID, Raman Sanyal ORCID

    Journal reference : Volume 28, 2014, pp. 1540-1558

    DOI : 10.1137/130944849

  • The PBW filtration and convex polytopes in type B

    Journal of Pure and Applied Algebra

    Authors : Teodor Backhaus, Deniz Kus ORCID

    Journal reference : Volume 223, 2018, pp. 245-276

    DOI : 10.1016/j.jpaa.2018.03.009

  • Two Enriched Poset Polytopes

    arXiv (Cornell University)

    Authors : Soichi Okada, Akiyoshi Tsuchiya ORCID

    Journal reference : Volume 28, 2023, pp. 257-282

    DOI : 10.1007/s00026-023-00679-7

  • Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

    Transactions of the American Mathematical Society Series B

    Authors : Naoki Fujita ORCID

    Journal reference : Volume 10, 2023, pp. 452-481

    DOI : 10.1090/btran/142
Preview
Loading PDF preview...