Abstract
We present a fast algorithm for computing the global crystal basis of the basic\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )\)-module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an th root of 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Olsson, J.B.: Partition identities with an application to group representation theory. J. Reine Angew. Math.413, 198–212 (1991)
Ariki, S.: On the decomposition numbers of the Hecke algebra ofG(m, 1,n). Preprint 1996
Bessenrodt, C., Olsson, J.B.: On Mullineux symbols. J. Comb. Theory Ser. A68, 340–360 (1994)
Curtis, C., Reiner, I.: Methods of representation theory with applications to finite groups and orders. Vol. 1, New York: Wiley, 1981
Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Paths, Maya diagrams, and representations of\(\widehat{\mathfrak{s}\mathfrak{l}}(r,C)\). Adv. Stud. Pure Math.19, 149–191 (1989)
Davies, B., Foda, O., Jimbo, M., Miwa, T., Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by vertex operators. Commun. Math. Phys.151, 89–153 (1993)
Dipper, R., James, G.: Representations of Hecke algebras of general linear groups. Proc. Lond. Math. Soc.52, 20–52 (1986)
Dipper, R., James, G.: Blocks and idempotents of Hecke algebras of general linear groups. Proc. Lond. Math. Soc.54, 57–82 (1987)
Du, J.: Canonical bases for irreducible representations of quantumGL n . Bull. Lond. Math. Soc.24, 325–334 (1992)
Du, J.: Canonical bases for irreducible representations of quantumGL n II. J. Lond. Math. Soc.51, 461–470 (1995)
Duchamp, G., Krob, D., Lascoux, A., Leclerc, B., Scharff, T., Thibon, J.-Y.: Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras. Publ. RIMS, Kyoto Univ.31, 179–201 (1995)
Ford, B., Kleshchev, A.S.: A proof of the Mullineux conjecture. Preprint 1994
Hayashi, T.:q-analogues of Clifford and Weyl algebras — Spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys.127, 129–144 (1990)
James, G.: The decomposition matrices ofGL n (q) forn≦10 Proc. Lond. Math. Soc.60, 225–265 (1990)
James, G., Kerber, A.: The representation theory of the symmetric group. Reading, MA: Addison-Wesley, 1981
James, G., Mathas, A.: Hecke algebras of typeA withq=−1. Preprint 1995
James, G., Mathas, A.: Aq-analogue of the Jantzen-Schaper theorem. Preprint 1995
Kac, V.: Infinite dimensional Lie algebras. Cambridge: Cambridge University Press, 1990
Kashiwara, M.: On crystal bases of theq-analogue of universal enveloping algebras. Duke Math. J.63, 465–516 (1991)
Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J.69, 455–485 (1993)
Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J.73, 383–413 (1994)
Kazhdan, D., Lusztig, G.: Affine Lie algebras and quantum groups. Int. Math. Res. Not.2, 21–29 (1991)
Kleshchev, A.S.: Branching rules for the modular representations of symmetric groups I. J. Algebra (to appear)
Kleshchev, A.S.: Branching rules for the modular representations of symmetric groups II. J. Reine. Angew. Math.459, 163–212 (1995)
Kleshchev, A.S.: Branching rules for the modular representations of symmetric groups III. Some corollaries and a problem of Mullineux. J. Lond. Math. Soc.2 (1995)
Lakshmibai, V.: Bases for quantum Demazure modules II. Proc. Symp. Pure. Math.56, part 2, 149–168 (1994)
Lascoux, A.: Leclerc, B., Thibon, J.-Y.: Ribbon tableaux, Hall-Littlewood functions and unipotent varieties. In: “Algebraic Combinatorics, Rencontre 12–16 March 1995, Saint-Nabor, Ottrott, France”, Université Louis Pasteur, IRMA, Strasbourg 1995, pp. 183–207
Lascoux, A.: Leclerc, B., Thibon, J.-Y.: Une conjecture pour le calcul des matrices de décomposition des algèbres de Hecke de typeA aux racines de l'unité. C.R. Acad. Sci. Paris321, 511–516 (1995)
Lascoux, A.: Leclerc, B., Thibon, J.-Y.: Crystal graphs andq-analogues of weight multiplicities for the root systemA n . Lett. Math. Phys.35, 359–374 (1995)
Leclerc, B., Thibon, J.-Y.: Canonical bases ofq-deformed Fock spaces. Internat. Math. Research Notices 9 (1996)
Lusztig, G.: Modular representations and quantum groups. Contemp Math.82, 59–77 (1989)
Macdonald, I.G.: Symmetric functions and Hall polynomials. 2nd edition. Oxford: Oxford Univ. Press 1995
Martin, S.: Schur algebras and representation theory. Cambridge: Cambridge University Press, 1993
Martin, S.: On the ordinary quiver of the principal block of certain symmetric groups. Quart. J. Math. Oxford Ser.40, 209–223 (1989)
Martin, S.: Ordinary quivers for symmetric groups. Quart. J. Math. Oxford Ser.41, 72–92 (1990)
Misra, K.C., Miwa, T.: Crystal base of the basic representation of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )\). Commun. Math. Phys.134, 79–88 (1990)
Mullineux, G.: Bijections ofp-regular partitions andp-modular irreducibles of the symmetric groups. J. Lond. Math.20, 60–66 (1979)
Richards, M.J.: Some decomposition numbers for Hecke algebras of general linear groups.
Robinson, G. De B.: Representation theory of the symmetric group. Edimburgh, 1961
Zelevinsky, A.: Representations of finite classical groups, a Hopf algebra approach. Lect. Notes in Math.869. Berlin-Heidelberg-New York: Springer, 1981
Supplementary reference
Grojnowski, I.: Representations of affine Hecke algebras (and affine quantumGL n ) at roots of unity. Internat. Math. Research Notices5, 215–217 (1994)
Author information
Authors and Affiliations
Additional information
Communicated by M. Jimbo
Partially supported by PRC Math-Info and EEC grant n0 ERBCHRXCT930400.
Rights and permissions
About this article
Cite this article
Lascoux, A., Leclerc, B. & Thibon, JY. Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Commun.Math. Phys. 181, 205–263 (1996). https://doi.org/10.1007/BF02101678
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02101678