Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 039, 13 pages      arXiv:1301.4237      https://doi.org/10.3842/SIGMA.2014.039
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Fusion Procedure for Cyclotomic Hecke Algebras

Oleg V. Ogievetsky a, b, c and Loïc Poulain d'Andecy d
a) Center of Theoretical Physics, Aix Marseille Université, CNRS, UMR 7332, 13288 Marseille, France
b) Université de Toulon, CNRS, UMR 7332, 83957 La Garde, France
c) On leave of absence from P.N. Lebedev Physical Institute, Leninsky Pr. 53, 117924 Moscow, Russia
d) Mathematics Laboratory of Versailles, LMV, CNRS UMR 8100, Versailles Saint-Quentin University, 45 avenue des Etas-Unis, 78035 Versailles Cedex, France

Received September 28, 2013, in final form March 29, 2014; Published online April 01, 2014

Abstract
A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

Key words: cyclotomic Hecke algebras; fusion formula; idempotents; Young tableaux; Jucys-Murphy elements; Schur element.

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References

  1. Ariki S., On the semi-simplicity of the Hecke algebra of $(\mathbb{Z}/r\mathbb{Z})\wr {\mathfrak S}_n$, J. Algebra 169 (1994), 216-225.
  2. Ariki S., Koike K., A Hecke algebra of $(\mathbb{Z}/r\mathbb{Z})\wr {\mathfrak S}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), 216-243.
  3. Broué M., Malle G., Zyklotomische Heckealgebren. Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 119-189.
  4. Cherednik I.V., A new interpretation of Gel'fand-Tzetlin bases, Duke Math. J. 54 (1987), 563-577.
  5. Chlouveraki M., Jacon N., Schur elements for the Ariki-Koike algebra and applications, J. Algebraic Combin. 35 (2012), 291-311, arXiv:1105.5910.
  6. Geck M., Iancu L., Malle G., Weights of Markov traces and generic degrees, Indag. Math. (N.S.) 11 (2000), 379-397.
  7. Isaev A.P., Molev A.I., Ogievetsky O.V., Idempotents for Birman-Murakami-Wenzl algebras and reflection equation, arXiv:1111.2502.
  8. Isaev A.P., Molev A.I., Os'kin A.F., On the idempotents of Hecke algebras, Lett. Math. Phys. 85 (2008), 79-90, arXiv:0804.4214.
  9. Isaev A.P., Ogievetsky O.V., On Baxterized solutions of reflection equation and integrable chain models, Nuclear Phys. B 760 (2007), 167-183, math-ph/0510078.
  10. Kulish P.P., Yang-Baxter equation and reflection equations in integrable models, in Low-Dimensional Models in Statistical Physics and Quantum Field Theory (Schladming, 1995), Lecture Notes in Phys., Vol. 469, Springer, Berlin, 1996, 125-144.
  11. Mathas A., Matrix units and generic degrees for the Ariki-Koike algebras, J. Algebra 281 (2004), 695-730, math.RT/0108164.
  12. Molev A.I., On the fusion procedure for the symmetric group, Rep. Math. Phys. 61 (2008), 181-188, math.RT/0612207.
  13. Ogievetsky O.V., Poulain d'Andecy L., On representations of cyclotomic Hecke algebras, Modern Phys. Lett. A 26 (2011), 795-803, arXiv:1012.5844.
  14. Ogievetsky O.V., Poulain d'Andecy L., Fusion formula for Coxeter groups of type B and complex reflection groups $G(m,1,n)$, arXiv:1111.6293.
  15. Ogievetsky O.V., Poulain d'Andecy L., Induced representations and traces for chains of affine and cyclotomic Hecke algebras, arXiv:1312.6980.

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