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Hecke algebras, difference operators, and quasi-symmetric functions. (English) Zbl 0990.05129
Quasi-symmetric functions and noncommutative symmetric functions are two recent generalizations of symmetric functions. The paper under review extends both of these concepts to Hall-Littlewood functions, defining quasi-symmetric Hall-Littlewood functions and noncommutative Hall-Littlewood functions. The development of these two functions parallels that of the ordinary Hall-Littlewood functions: the author first considers that Hall-Littlewood functions can be built up using Hecke algebras and Demazure’s character formula [see G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. Scharf and J.-Y. Thibon, Euler-Poincaré characteristic and polynomial representations of Iwashori-Hecke algebras, Publ. Res. Inst. Math. Sci. 31, No. 2, 179-201 (1995; Zbl 0835.05085)], and then extends these techniques to produce quasi-symmetric Hall-Littlewood functions. The author finally employs duality to derive the noncommutative Hall-Littlewood functions.

MSC:
05E05 Symmetric functions and generalizations
20C08 Hecke algebras and their representations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
39A70 Difference operators
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Citations:
Zbl 0835.05085
Software:
NCSF
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Full Text: DOI
References:
[1] Bernstein, I.N.; Gelfand, I.M.; Gelfand, S.I., Schubert cells and the cohomology of the space G/P, Russian math. surveys, 28, 1-26, (1973) · Zbl 0289.57024
[2] Bouwknegt, P.; Pilch, K., The deformed Virasoro algebra at roots of unity, Commun. math. phys, 196, 249-288, (1998) · Zbl 0974.17014
[3] Carter, R.W., Representation theory of the 0-Hecke algebra, J. algebra, 15, 89-103, (1986) · Zbl 0624.20007
[4] Cherednik, I., A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. math., 106, 411-432, (1991) · Zbl 0725.20012
[5] Demazure, M., Une formule des caractères, Bull. sci. math., 98, 163-172, (1974) · Zbl 0365.17005
[6] Dipper, R.; Donkin, S., Quantum GL_{n}, Proc. London math. soc., 63, 165-211, (1991) · Zbl 0734.20018
[7] Duchamp, G.; Krob, D.; Lascoux, A.; Leclerc, B.; Scharf, T.; Thibon, J.-Y., Euler- Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras, Publ. res. inst. math. sci., 31, 179-201, (1995) · Zbl 0835.05085
[8] Duchamp, G.; Krob, D.; Leclerc, B.; Thibon, J.-Y., Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à q=0, C. R. acad. sci. Paris, 322, 107-112, (1996) · Zbl 0839.20017
[9] Foulkes, H.O., A survey of some combinatorial aspects of symmetric functions, Permutations, (1974), Gauthier-Villars Paris · Zbl 0282.05004
[10] Garsia, A.M.; Haiman, M., A graded representation model for Macdonald’s polynomials, Proc. nat. acad. sci. U.S.A., 90, 3607-3610, (1993) · Zbl 0831.05062
[11] Gelfand, I.M.; Krob, D.; Leclerc, B.; Lascoux, A.; Retakh, V.S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 218-348, (1995) · Zbl 0831.05063
[12] Gessel, I., Multipartite P-partitions and inner products of skew Schur functions, (), 289-301
[13] Green, J.A., The characters of the finite general linear groups, Trans. amer. math. soc., 80, 402-449, (1955) · Zbl 0068.25605
[14] Hall, P., The algebra of partitions, (), 70-84
[15] F. Hivert, Affine Hecke algebra and quasi-symmetric functions, preprint, 1998.
[16] Hotta, R.; Springer, T.A., A specialization theorem for certain Weyl group representations and application to the Green polynomials of unitary groups, Invent. math., 41, 113-140, (1977) · Zbl 0389.20037
[17] Johnsen, K., On a forgotten note by ernst Steinitz on the theory of abelian groups, Bull. London math. soc., 14, 353-355, (1982) · Zbl 0491.20042
[18] Kirillov, A.N.; Reshetikhin, N.Yu., Bethe ansatz and the combinatorics of Young tableaux, J. sov. math., 41, 925-955, (1988) · Zbl 0639.20029
[19] Klyachko, A.A., Lie element in the tensor algebra, Siberian math. J., 15, 1296-1304, (1974) · Zbl 0315.15015
[20] Krob, D.; Thibon, J.-Y., Noncommutative symmetric functions. IV. quantum linear groups and Hecke algebras at q=0, J. algebraic combin., 6, 339-376, (1997) · Zbl 0881.05120
[21] Krob, D.; Thibon, J.-Y., A crystalizable version of U_{q}(\(G\)ln), ()
[22] D. Krob, and, J.-Y. Thibon, Noncommutative symmetric functions. V. A degenerate version of U_{q}(glN), preprint, 1997.
[23] Lascoux, A., Cyclic permutations on words, tableaux and harmonic polynomials, Proc. conf. on algebraic groups, Hyderabad, 1989, (1991), Manoj Prakashar Madras, p. 323-347 · Zbl 0823.20012
[24] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Fonctions de Hall-Littlewood et polynômes de kostka-foulkes aux racines de l’unité, C. R. acad. sci. Paris, 316, 1-6, (1993) · Zbl 0769.05095
[25] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Green polynomials and Hall-Littlewood functions at roots of unity, European J. combin., 15, 173-180, (1994) · Zbl 0789.05093
[26] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Polynômes de kostka-foulkes et graphes cristallins des groupes quantiques de type A_{n}, C. R. acad. sci. Paris, 320, 131-134, (1995) · Zbl 0854.17013
[27] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. math. phys., 38, 1041-1068, (1997) · Zbl 0869.05068
[28] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. math. phys., 181, 205-263, (1997) · Zbl 0874.17009
[29] Lascoux, A.; Schützenberger, M.P., Le monoode plaxique, Quad. ric. sci., 109, 129-156, (1981)
[30] Lascoux, A.; Schützenberger, M.P., Symmetry and flag manifold, Invariant theory, 996, 118-144, (1983)
[31] Lascoux, A.; Schützenberger, M.P., Key and standard bases, Invariant theory and tableaux, IMA maths. appl., 19, (1990), Springer-Verlag New York, p. 125-144 · Zbl 0815.20013
[32] Lascoux, A.; Schützenberger, M.P., Symmetrization operators on polynomial rings, Funct. anal. appl., 21, 77-78, (1987) · Zbl 0659.13008
[33] Leclerc, B.; Thibon, J.-Y., Canonical bases of q-deformed Fock spaces, Internat. math. res. notices, 9, 447-456, (1996) · Zbl 0863.17013
[34] P. Littelman, Crystal Graphs and Young tableaux, J. Algebra, in press.
[35] Littlewood, D.E., On certain symmetric functions, Proc. London math. soc., 43, 485-498, (1961) · Zbl 0099.25102
[36] Lusztig, G., Green polynomials and singularities of unipotent classes, Adv. math., 42, 169-257, (1981) · Zbl 0473.20029
[37] Lusztig, G., Equivariant K-theory and representations of Hecke algebras, Proc amer. math. soc., 94, 337-342, (1985) · Zbl 0571.22014
[38] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Clarendon Oxford · Zbl 0487.20007
[39] Macdonald, I.G., Note on Schubert polynomials, (1991), Publications du LACIM Montreal · Zbl 0784.05061
[40] MacMahon, P.A., Combinatorial analysis, (1915), Cambridge Univ. Press Cambridge · JFM 46.0118.07
[41] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and Solomon descent algebra, J. algebra, 177, 967-982, (1995) · Zbl 0838.05100
[42] Nakayashiki, A.; Yamada, Y., Kotska polynomials and energy functions in solvable lattice models, Selecta math., 3, 547-599, (1997) · Zbl 0915.17016
[43] Novelli, J.-C., On the hypoplactic monoid, () · Zbl 0960.05106
[44] Schur, I., Über die darstellung der symmetrischen und der alternieren den gruppe durch gebrochene lineare substitutionen, Crelle’s J., 139, 155-250, (1911) · JFM 42.0154.02
[45] Steinberg, E., A geometric approach to the representations of the full linear group over a Galois field, Trans. amer. math. soc., 71, 274-282, (1951) · Zbl 0045.30201
[46] Steinitz, E., Zur theorie der Abel’schen gruppen, Jahresber. Deutsch. math.-verein., 9, 80-85, (1901) · JFM 32.0149.02
[47] Takeuchi, M., A two-parameter quantization of GL(n), Proc. Japan acad. ser. A, 66, 112-114, (1990) · Zbl 0723.17012
[48] Ung, B.C.V., NCSF, a Maple package for noncommutative symmetric functions, Maple tech. news., 3, 24-29, (1996)
[49] B. C. V. Ung, and, S. Veigneau, ACE une environnement en combinatoire algébrique, in, Proc. of the 7th Conf. Formal Power Series and Algebraic Combinatorics, 1995.
[50] Yang, C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. rev. lett., 19, 1312-1325, (1967) · Zbl 0152.46301
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