Skip to main content
SpringerLink
Log in

Dual canonical bases, quantum shuffles and q-characters

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

Rosso and Green have shown how to embed the positive part U q () of a quantum enveloping algebra U q () in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis B * of U q () under this embedding Φ. This is motivated by the fact that when is of type A r , the elements of Φ(B *) are q-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups GL(m) over a p-adic field.

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

    MathSciNet  MATH  Google Scholar 

  2. Berenstein, A., Zelevinsky, A.: String bases for quantum groups of type A r . Advances~in Soviet Math. (Gelfand’s seminar) 16, 51–89 (1993)

    MATH  Google Scholar 

  3. Berenstein, A., Zelevinsky, A.: Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143, 77–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourbaki, N.: Groupes et algèbres de Lie, Chapitre 6, Hermann 1968

  5. Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra l(m|n). J. Am. Math. Soc. 16, 185–231 (2003)

    Google Scholar 

  6. Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra (n). Preprint, math.RT/0207024 (Advances in Math. to appear)

  7. Brundan, J., Kleshchev, A.: Hecke-Clifford superalgebras, crystals of type A 2l (2) and modular branching rules for \(\widehat{S}_n\). Representation theory 5, 317–403 (2001)

    Article  MATH  Google Scholar 

  8. Caldero, P.: Adapted algebras for the Berenstein-Zelevinsky conjecture. Transform. Groups 8, 37–50 (2003)

    MATH  Google Scholar 

  9. De Concini, C., Procesi, C.: Quantum groups. In: D-modules, representation theory, and quantum groups (Venezia 1992), Springer Lecture Notes in Math. 1565, 1993

  10. Chari, V., Xi, N.: Monomial bases of quantized enveloping algebras. In: Recent developments in quantum affine algebras and related topics. A.M.S. Contemp. Math. 248, 69–81 (1999)

    MATH  Google Scholar 

  11. Frenkel, E., Mukhin, E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Comm. Math. Phys. 216, 23–57 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of -algebras. In: ‘Recent Developments in quantum affine algebras and related topics’. A.M.S. Contemp. Math. 248, 163–205

    Google Scholar 

  13. Green, J.A.: Quantum groups, Hall algebras and quantum shuffles. In: Finite reductive groups (Luminy 1994), 273–290, Birkhäuser Prog. Math. 141, 1997

    Google Scholar 

  14. Grojnowski, I.: Affine controls the modular representation theory of the symmetric group and related Hecke algebras. Preprint 1999

  15. Grojnowski, I., Vazirani, M.: Strong multiplicity one theorem for affine Hecke algebras of type A. Transform. Groups 6, 143–155 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Jones, A., Nazarov, M.: Affine Sergeev algebra and q-analogues of the Young symmetrizers for the projective representations of the symmetric group. Proc. London Math. Soc. 78, 481–512 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Lalonde, P., Ram, A.: Standard Lyndon bases of Lie algebras and enveloping algebras. Trans. Am. Math. Soc. 347, 1821–1830 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Lascoux, A., Leclerc, B., Thibon, J.-Y.: Une conjecture pour le calcul des matrices de décomposition des algèbres de Hecke aux racines de l’unité. C.R. Acad. Sci. Paris 321, 511–516 (1995)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Leclerc, B.: Imaginary vectors in the dual canonical basis of U q (). Transform. Groups 8, 95–104 (2003)

    Google Scholar 

  22. Leclerc, B., Nazarov, M., Thibon, J.-Y.: Induced representations of affine Hecke algebras and canonical bases of quantum groups. In: Studies in memory of Issai Schur. Progress in Mathematics 210, Birkhäuser, 2003

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

    Article  MATH  Google Scholar 

  24. Leclerc, B., Thibon, J.-Y.: q-Deformed Fock spaces and modular representations of spin symmetric groups. J. Phys. A 30, 6163–6176 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Levendorskii, S.Z., Soibelman, S.: Algebras of functions on compact quantum groups, Schubert cells and quantum tori. Comm. Math. Phys. 139, 141–170 (1991)

    MathSciNet  Google Scholar 

  26. Lothaire, M.: Combinatorics on Words. Readings, Massachusetts, 1983

  27. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)

    Google Scholar 

  28. Lusztig, G.: Introduction to quantum groups. Birkhäuser Prog. Math. 110, 1993

    Google Scholar 

  29. Lusztig, G.: Total positivity in reductive groups. In: Lie Theory and Geometry: in Honor of Bertram Kostant. Progress in Math. 123, Birkhäuser, 1994

  30. Melançon, G.: Combinatorics of Hall trees and Hall words. J. Combinat. Theory A 59, 285–308 (1992)

    MathSciNet  Google Scholar 

  31. Nakajima, H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Preprint, math.QA/0105173 (Annals of Math. to appear)

  32. Olshanski, G.: Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra. Lett. Math. Phys. 24, 93–102 (1992)

    MathSciNet  Google Scholar 

  33. Papi, P.: A characterization of a special ordering in a root system. Proc. Am. Math. Soc. 120, 661–665 (1994)

    MathSciNet  MATH  Google Scholar 

  34. Reineke, M.: Feigin’s map and monomial bases for quantized enveloping algebras. Math. Z. 237, 639–667 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Retakh, V., Zelevinsky, A.: The base affine space and canonical bases in irreducible representations of the group Sp(4). Soviet Math. Dokl. 37, 618–622 (1988)

    MATH  Google Scholar 

  36. Reutenauer, C.: Free Lie algebras. Oxford University Press, 1993

  37. Ringel, C.M.: PBW-bases of quantum groups. J. Reine Angew. Math. 470, 51–88 (1996)

    MathSciNet  MATH  Google Scholar 

  38. Rogawski, J.D.: On modules over the Hecke algebra of a p-adic group. Invent. Math. 79, 443–465 (1985)

    MathSciNet  MATH  Google Scholar 

  39. Rosso, M.: Groupes quantiques et algèbres de battage quantiques. C. R. Acad. Sci. Paris 320, 145–148 (1995)

    MathSciNet  MATH  Google Scholar 

  40. Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133, 399–416 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rosso, M.: Lyndon bases and the multiplicative formula for R-matrices. Preprint 2002

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

    MathSciNet  MATH  Google Scholar 

  43. Zelevinsky, A.: Induced representations of reductive p-adic groups II. Ann. Sci. E.N.S. 13, 165–210 (1980)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Caen, Campus II, Bld Maréchal Juin, BP 5186, 14032, Caen cedex, France

    Bernard Leclerc

Authors
  1. Bernard Leclerc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard Leclerc.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leclerc, B. Dual canonical bases, quantum shuffles and q-characters. Math. Z. 246, 691–732 (2004). https://doi.org/10.1007/s00209-003-0609-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0609-9

Keywords

Search

Navigation