×

zbMATH — the first resource for mathematics

A Frobenius formula for the characters of the Hecke algebras. (English) Zbl 0758.05099
In this paper, characters for representations of the Hecke algebra of type \(A_ n\) are studied. The natural action of this Hecke algebra on the tensor products of the standard representation of the quantum group \(U_ q(sl(n))\) is used. By rewriting this action in an appropriate form, a Frobenius type formula for the characters is deduced. Using this formula, a combinatorial rule is given to compute the irreducible characters of the Hecke algebra, and some tables are given. Connections with Hall-Littlewood symmetric functions and Kronecker products of symmetric groups are pointed out.

MSC:
05E10 Combinatorial aspects of representation theory
20C15 Ordinary representations and characters
17B37 Quantum groups (quantized enveloping algebras) and related deformations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Bou] Bourbaki, N.: Groupes et Algèbres de Lie, Chapters 4-6. Paris: Hermann 1960
[2] [Dr] Drinfel’d, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Berkeley, 1986, pp. 798-820, Providence, RI: Am. Math. Soc. 1987
[3] [FYHLMO] Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millet, K., Oeneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc.12, 239-246 (1985) · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3
[4] [Fr] Frobenius, F.G.: Über die Charactere der symmetrischen Gruppe. Sitzungsber. K. Preuss. Akad. Wisse. Berlin, 516-534 (1900); reprinted in: Gessamelte Abhandlungen 3, pp. 148-166, Berlin Heidelberg New York: Springer 1973 · JFM 31.0129.02
[5] [G-R] Garsia, A.M., Remmel, J.: Shuffles of permutations and the Kronecker product. Graphs Comb.1, 217-263 (1985) · Zbl 0588.05005 · doi:10.1007/BF02582950
[6] [Gy] Gyoja, A.: Aq-analogue of Young symmetrizer. Osaka J. Math.23, 841-852 (1986) · Zbl 0644.20012
[7] [H] Hoefsmit, P.N.: Representations of Hecke algebras of finite groups with BN pairs of classical type. Thesis. University of British Columbia: 1974
[8] [Ji] Jimbo, M.: Aq-analogue of \(U(\mathfrak{g}l(N + 1))\) , Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys.11, 247-252 (1986) · Zbl 0602.17005 · doi:10.1007/BF00400222
[9] [Ji2] Jimbo, M.: Introduction to the Yang-Baxter equation. (Preprint 1989) · Zbl 0697.35131
[10] [Jo] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335-388 (1987) · Zbl 0631.57005 · doi:10.2307/1971403
[11] [K] Kerov, S.V.: Generalized Hall-Littlewood symmetric functions and orthogonal polynomials. (Preprint 1990) · Zbl 0760.05089
[12] [KW1] King, R.C., Wybourne, B.G.: Characters of Hecke algebrasH n (q) of typeA n?1 .J. Phys. A. (to appear)
[13] [KW2] King, R.C., Wybourne, B.G.: Representations and traces of the Hecke algebrasH n (q) of typeA n?1 .(Preprint 1990)
[14] [Mac] Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford: Clarendon Press 1979 · Zbl 0487.20007
[15] [O] Ocneanu, A.: A polynomial invariant for knots: a combinatorial and an algebraic approach (Preprint) (see [FYHLMO]) for a shortened version
[16] [Sc1] Schur, I.: Über eine Klasse von Matrizen, die sich einer gegeben Matrix zuordnen lassen. Dissertation, 1901; reprinted in: Gessamelte Abhandlungen 1, pp. 1-72. Berlin Heidelberg New York: Springer 1973
[17] [Sc2] Schur, I.: Über die rationalen Darstellungen der allgemeinen linearen Gruppe. Sitzungsber. K. Preuss. Akad. Wiss. Berlin, 58-75 (1927). Reprinted in: Gessamelte Abhandlungen 3, pp. 68-85. Berlin Heidelberg New York: Springer 1973 · JFM 53.0108.05
[18] [VK1] Vershik, A.M., Kerov, S.V.: Characters, factor representations andK-functor of the infinite symmetric group. Proc. Int. Conf. on Oper. Algebras and Group Repres. vol. II Romania 1980. (Monogr. Stud. Math., vol. 13, pp. 23-34) 1984
[19] [VK2] Vershik, A.M., Kerov, S.V.: Characters and realizations of representations of an infinite-dimensional Hecke algebra, and knot invariants. Sov. Math., Dokl.38, 134-137 (1989) · Zbl 0716.20008
[20] [Wz1] Wenzl, H.: Hecke algebras of typeA n and subfactors. Invent. Math.92, 349-383 (1988) · Zbl 0663.46055 · doi:10.1007/BF01404457
[21] [Wz2] Wenzl, H.: Quantum groups and subfactors of typeB, C andD. Commun. Math. Phys.133, 383-432 (1990) · Zbl 0744.17021 · doi:10.1007/BF02097374
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.