Skip to main content
SpringerLink
Log in

Symmetric Functions and the Fock Space

  • Conference paper
Book cover Symmetric Functions 2001: Surveys of Developments and Perspectives

Part of the book series: NATO Science Series ((NAII,volume 74))

Abstract

We review the definition, calculation and properties of the canonical bases of the Fock space representation of EquationSource$$ U_q (\widehat{sl}_n ) $$. We emphasize the close connection with the theory of symmetric functions (plethysm, Hall-Littlewood functions, Macdonald polynomials)

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Ariki, On the decomposition numbers of the Hecke algebra of G(n, 1, m), J. Math. Kyoto Univ. 36 (1996), 789–808.

    MathSciNet  MATH  Google Scholar 

  2. J. Beck, I. B. Prenkel, N. Jing, Canonical basis and Macdonald polynomials, Adv. Math. 140 (1998), 95–127.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Carré, B. Leclerc, Splitting the square of a Schur function into its symmetric and antisymmetric parts, J. Algebraic Combinatorics 4 (1995), 201–231.

    Article  MATH  Google Scholar 

  4. J. Chuang, R. Kessar, Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture, Bull. London Math. Soc. 34 (2002), 174–184.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Chuang, K. M. Tan, Canonical basis of the basic EquationSource$$ U_q (\widehat{sl}_n ) $$ -module, Preprint 2001.

    Google Scholar 

  6. V. V. Deodhar, On some geometric aspects of Bruhat orderings II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra, 111 (1987), 483–506.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.

    Google Scholar 

  8. K. Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de-Vries type equation, Lect. Notes Math., 933 (1982), 71–110.

    Article  MathSciNet  Google Scholar 

  10. I. B. Prenkel, Representations of affine Kac-Moody algebras and dual resonance models, Lectures in Appl. Math., 21 (1985), 325–353.

    Google Scholar 

  11. F. Goodman, H. Wenzl, Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials, Internat. Math. Res. Notices, 5 (1999), 251–275.

    Article  MathSciNet  Google Scholar 

  12. T. Hayashi, q-analogues of Clifford and Weyl algebras — spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129–144.

    Article  MathSciNet  Google Scholar 

  13. A. Hida, H. Miyachi, Some blocks of unite general linear groups in non defining characteristic, (2000), Preprint.

    Google Scholar 

  14. G. James, The decomposition matrices of GL n(q)for n ⩽ 10, Proc. London Math. Soc., 60 (1990), 225–265.

    Article  MathSciNet  MATH  Google Scholar 

  15. G.D. James, A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics 16, Addison-Wesley, 1981.

    Google Scholar 

  16. G. James, A. Mathas, Hecke algebras of type A at q =-1, J. Algebra 184 (1996), 102–158.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Jimbo, A q-difference analogue of U(g)and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Jimbo, T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS, Kyoto Univ. 19 (1983), 943–1001.

    Article  MathSciNet  MATH  Google Scholar 

  19. V. G. Kac, Infinite dimensional Lie algebras, 3rd Ed. Cambridge University Press, 1990.

    Google Scholar 

  20. M. Kashiwara, On level zero representations of quantized affine algebras, 2000, math.QA/0010293.

    Google Scholar 

  21. M. Kashiwara, T. Miwa, E. Stern, Decomposition of q-deformed Fock spaces, Selecta Math. 1 (1995), 787–805.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Kashiwara, T. Miwa, J.-U. Petersen, C. M. Yung, Perfect crystals and q-deformed Fock spaces, Selecta Math. 2 (1996), 415–499.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Kashiwara, T. Tanisaki, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, 1999, math.RT/9908153.

    Google Scholar 

  24. A. Lascoux, B. Leclerc, J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Commun. Math. Phys. 181 (1996), 205–263.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997), 1041–1068.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Leclerc, Decomposition numbers and canonical bases, Algebras and representation theory, 3 (2000), 277–287.

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Leclerc, H. Miyachi, Some closed formulas for canonical bases of Fock spaces, 2001, math.QA/0104107.

    Google Scholar 

  28. B. Leclerc, J.-Y. Thibon, Canonical bases of q-deformed Fock spaces, Int. Math. Res. Notices, 9 (1996), 447–456.

    Article  MathSciNet  Google Scholar 

  29. B. Leclerc, J.-Y. Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, in Combinatorial methods in representation theory, Ed. M. Kashiwara et al., Adv. Stud. Pure Math. 28 (2000), 155–220.

    Google Scholar 

  30. D. E. Littlewood, Modular representations of symmetric groups, Proc. Roy. Soc. 209 (1951), 333–353.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Lusztig, Green polynomials and singularities of unipotent classes, Advances in Math. 42 (1981), 169–178.

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, Analyse et topologie sur les espaces singuliers (II-III), Astérisque 101-102 (1983), 208–227.

    MathSciNet  Google Scholar 

  33. G. Lusztig, Introduction to quantum groups, Progress in Math. 110, Birkhauser 1993.

    Google Scholar 

  34. K.C. Misra, T. Miwa, Crystal base of the basic representation of EquationSource$$ U_q (\widehat{sl}_n ) $$, Commun. Math. Phys.134} (1990}), 79–88

    Article  MathSciNet  MATH  Google Scholar 

  35. LG. Macdonald, Symmetric functions and Hall polynomials, Oxford U. Press, 1995.

    Google Scholar 

  36. O. Schiffmann, The Hall algebra of the cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 8 (2000), 413–440.

    Article  MathSciNet  Google Scholar 

  37. W. Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory 1 (1997), 37–68 (english 83-114).

    Article  MathSciNet  MATH  Google Scholar 

  38. K. Takemura, D. Uglov, Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type glN, Publ. RIMS, Kyoto Univ. 35 (1999), 407–450.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Uglov, Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physical Combinatorics Ed. M. Kashiwara, T. Miwa, Progress in Math. 191, Birkhauser 2000, 249–299.

    Google Scholar 

  40. M. A. A. van Leeuwen, Some bijective correspondences involving domino tableaux, Electron. J. Comb. 7 (2000), R35, 25 p.

    Google Scholar 

  41. M. Varagnolo, E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Caen, France

    Bernard Leclerc

Authors
  1. Bernard Leclerc
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Sergey Fomin

Additional information

Dedicated to Denis Uglov (1968–1999)

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media Dordrecht

About this paper

Cite this paper

Leclerc, B. (2002). Symmetric Functions and the Fock Space. In: Fomin, S. (eds) Symmetric Functions 2001: Surveys of Developments and Perspectives. NATO Science Series, vol 74. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0524-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-0524-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0774-3

  • Online ISBN: 978-94-010-0524-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation