Skip to main content
SpringerLink
Log in

New Singular Gelfand–Tsetlin \({\mathfrak{gl}(n)}\)-Modules of Index 2

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Singular Gelfand–Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand–Tsetlin modules for arbitrary singular character of index 2. Explicit bases of derivative tableaux and the action of the generators of \({\mathfrak{gl}(n)}\) are given for these modules. Our construction leads to new families of irreducible Gelfand–Tsetlin modules and also provides tableaux bases for some simple Verma modules.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Drozd Y., Ovsienko S., Futorny V.: Irreducible weighted \({\mathfrak{sl}(3)}\)-modules. Funksionalnyi Analiz i Ego Prilozheniya 23, 57–58 (1989)

    MathSciNet  Google Scholar 

  2. Drozd Y., Ovsienko S., Futorny V.: Harish-Chandra subalgebras and Gelfand–Zetlin modules. Math. Phys. Sci. 424, 72–89 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Fernando S.: Lie algebra modules with finite dimensional weight spaces I. Trans. Am. Math. Soc. 322, 757–781 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Futorny V., Ovsienko S.: Fibers of characters in Gelfand–Tsetlin categories. Trans. Am. Math. Soc. 366, 4173–4208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Futorny, V., Grantcharov, D., Ramirez, L.E.: Irreducible generic Gelfand–Tsetlin modules of \({\mathfrak{gl}(n)}\). Symmetry, Integrability and Geometry: Methods and Applications, vol. 018 (2015)

  6. Futorny V., Grantcharov D., Ramirez L.E.: Singular Gelfand–Tsetlin modules for \({\mathfrak{gl}(n)}\). Adv. Math. 290, 453–482 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Futorny, V., Grantcharov, D., Ramirez, L.E.: Classification of irreducible Gelfand–Tsetlin modules for \({\mathfrak{sl}(3)}\). In: Progress

  8. Futorny, V., Grantcharov, D., Ramirez, L.E.: Drinfeld category and the classification of singular Gelfand–Tsetlin \({\mathfrak{gl}_n}\) -modules. Inter. Math. Res. Not. doi:10.1093/imrn/rnx159

  9. Gelfand, I., Graev, M.: Finite-dimensional irreducible representations of the unitary and complete linear group and special functions associated with them. Izv. Akad. Nauk SSSR Ser. Mat. 29, 1329–1356 (1965) (Russian)

  10. Gelfand I., Tsetlin M.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950)

    MathSciNet  Google Scholar 

  11. Gomez, C., Ramirez, L.E.: Families of irreducible singular Gelfand–Tsetlin modules of \({\mathfrak{gl}(n)}\). arXiv:1612.00636

  12. Graev M.: Infinite-dimensional representations of the Lie algebra \({\mathfrak{gl}(n, \mathbb{C})}\) related to complex analogs of the Gelfand–Tsetlin patterns and general hypergeometric functions on the Lie group \({GL(n, \mathbb{C})}\). Acta Appl. Math. 81, 93–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Graev M.: A continuous analogue of Gelfand–Tsetlin schemes and a realization of the principal series of irreducible unitary representations of the group \({GL(n,\mathbb{C})}\) in the space of functions on the manifold of these schemes. Dokl. Akad. Nauk. 412(2), 154–158 (2007)

    MathSciNet  Google Scholar 

  14. Guillemin V., Sternberg S.: The Gelfand–Tsetlin system and quantization of the complex flag manifolds. J. Funct. Anal. 52(1), 106–128 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kostant, B., Wallach, N.: Gelfand–Zeitlin theory from the perspective of classical mechanics I. In: Studies in Lie Theory Dedicated to A. Joseph on his Sixtieth Birthday. Progress in Mathematics, vol. 243, pp. 319–364 (2006)

  16. Kostant, B., Wallach, N.: Gelfand-Zeitlin theory from the perspective of classical mechanics II. In The Unity of Mathematics In Honor of the Ninetieth Birthday of I. M. Gelfand. Progress in Mathematics, vol. 244, pp. 387–420 (2006)

  17. Lemire F., Patera J.: Formal analytic continuation of Gelfand finite-dimensional representations of \({GL(n,\mathbb{C})}\). J. Math. Phys. 20, 820–829 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Lemire F., Patera J.: Gelfand representations of \({sl(n,\mathbb{C})}\). Algebras Groups Geom. 2, 14–166 (1985)

    MathSciNet  Google Scholar 

  19. Mathieu O.: Classification of irreducible weight modules. Ann. Inst. Fourier 50, 537–592 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mazorchuk V.: Tableaux realization of generalized Verma modules. Can. J. Math. 50, 816–828 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mazorchuk, V.: On Categories of Gelfand–Zetlin Modules, Noncommutative Structures in Mathematics and Physics. Kluwer Academic Publishers, Dordrecht, pp. 299–307 (2001)

  22. Molev, A.: Gelfand–Tsetlin bases for classical Lie algebras. In: Hazewinkel, M. (eds.) Handbook of Algebra, vol. 4. Elsevier, Amsterdam, pp. 109–170 (2006)

  23. Ovsienko, S.: Finiteness statements for Gelfand–Zetlin modules. In: Third International Algebraic Conference in the Ukraine (Ukrainian). Natsional. Akad. Nauk Ukrainy, Inst. Mat., Kiev, pp. 323–338 (2002)

  24. Ramirez L.E.: Combinatorics of irreducible Gelfand–Tsetlin \({\mathfrak{sl}(3)}\)-modules. Algebra Discrete Math. 14, 276–296 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Vinberg E.: On certain commutative subalgebras of a universal enveloping algebra. Math. USSR Izvestiya 36, 1–22 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Zhelobenko, D.: Compact Lie groups and their representations. Transl. Math. Monographs, AMS, vol. 40 (1974)

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil

    Vyacheslav Futorny

  2. University of Texas at Arlington, Arlington, TX, 76019, USA

    Dimitar Grantcharov

  3. Universidade Federal do ABC, Santo André, SP, Brazil

    Luis Enrique Ramirez

Authors
  1. Vyacheslav Futorny
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitar Grantcharov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luis Enrique Ramirez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dimitar Grantcharov.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Futorny, V., Grantcharov, D. & Ramirez, L.E. New Singular Gelfand–Tsetlin \({\mathfrak{gl}(n)}\)-Modules of Index 2. Commun. Math. Phys. 355, 1209–1241 (2017). https://doi.org/10.1007/s00220-017-2967-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2967-x

Search

Navigation