ELibM Journals • ELibM Home • EMIS Home • EMIS Mirrors

  EMIS Electronic Library of Mathematics (ELibM)
The Open Access Repository of Mathematics
  EMIS ELibM Electronic Journals

JOURNAL OF
ALGEBRAIC
COMBINATORICS

  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Canonical bases of higher-level q -deformed Fock spaces

Xavier Yvonne
Université Lyon I Institut Camille Jordan (Mathématiques) 43 Bd du 11 Novembre 1918 69622 Villeurbanne Cedex France

DOI: 10.1007/s10801-007-0062-7

Abstract

We show that the transition matrices between the standard and the canonical bases of infinitely many weight subspaces of the higher-level q-deformed Fock spaces are equal.

Pages: 383–414

Full Text: PDF

References

1. Ariki, S. (1996). On the decomposition numbers of the Hecke algebra of G(m, 1, n). J. Math. Kyoto Univ., 36(4), 789-808.
2. Dipper, R., James, G., & Mathas, A. (1998). Cyclotomic q-Schur algebras. Math. Z., 229, 385-416. (1)
3. Foda, O., Leclerc, B., Okado, M., Thibon, J.-Y., & Welsh, T. (1999). Branching functions of An - 1 and Jantzen-Seitz problem for Ariki-Koike algebras. Adv. Math., 141(2), 322-365.
4. Hayashi, T. (1990). q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys., 127(1), 129-144. J Algebr Comb (2007) 26: 383-414
5. Jacon, N. (2004). On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity. J. Math. Kyoto Univ., 44(4), 729-767.
6. Jimbo, M., Misra, K., Miwa, T., & Okado, M. (1991). Combinatorics of representations of Uq (sln) at q =
0. Commun. Math. Phys., 136(3), 543-566.
7. Kac, V. G. (1990). Infinite dimensional Lie algebras (3rd edn.). Cambridge: Cambridge University Press.
8. Kashiwara, M. (1993). Global crystal bases of quantum groups. Duke Math. J., 69, 455-485.
9. Kashiwara, M. (1993). The crystal base and Littelmann's refined Demazure character formula. Duke Math. J., 71, 839-958.
10. Kashiwara, M., Miwa, T., & Stern, E. (1995). Decomposition of q-deformed Fock spaces. Sel. Math., 1, 787-805.
11. Kashiwara, M., & Tanisaki, T. (2002). Parabolic Kazhdan-Lusztig polynomials and Schubert varieties. J. Algebra, 249, 306-325.
12. Lascoux, A., Leclerc, B., & Thibon, J.-Y. (1996). Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Commun. Math. Phys., 181(1), 205-263.
13. Leclerc, B., & Miyachi, H. (2002). Some closed formulas for canonical bases of Fock spaces. Represent. Theory, 6, 290-312.
14. Leclerc, B., & Thibon, J.-Y. (1996). Canonical bases of q-deformed Fock spaces. Intern. Math. Res. Not., 9, 447-456.
15. Leclerc, B., & Thibon, J.-Y. (2000). Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials. Comb. Methods Represent. Theory Adv. Stud. Pure Math., 28, 155-220.
16. Macdonald, I. G. (1990). Symmetric functions and Hall polynomials (2nd edn.). Oxford Science Publications. London: Oxford University Press.
17. Misra, K. C., & Miwa, T. (1990). Crystal base for the basic representation of Uq (sln). Commun. Math. Phys., 134(1), 79-88.
18. Scopes, J. (1991). Cartan matrices and Morita equivalence for blocks of the symmetric groups. J. Algebra, 142(2), 441-455.
19. Uglov, D. (2000). Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials. In M. Kashiwara, T. Miwa (Eds.), Physical combinatorics. Progress in math. (Vol. 191). Birkhäuser, Basel, math.QA/9905196 (1999).
20. Varagnolo, M., & Vasserot, E. (1999). On the decomposition matrices of the quantized Schur algebra. Duke Math. J., 100, 267-297.
21. Yvonne, X. (2005). Bases canoniques d'espaces de Fock de niveau supérieur. Thèse de l'Université de Caen.
22. Yvonne, X. (2006). A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras. J. Algebra, 304, 419-456.
23. Yvonne, X. (2007). An algorithm for computing the canonical bases of higher-level q-deformed Fock spaces. J. Algebra, 309, 760-785.




© 1992–2009 Journal of Algebraic Combinatorics
© 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition