Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Fusion procedure for Coxeter groups of type $ B$ and complex reflection groups $ G(m,1,n)$


Authors: O. V. Ogievetsky and L. Poulain d’Andecy
Journal: Proc. Amer. Math. Soc. 142 (2014), 2929-2941
MSC (2010): Primary 20C05, 20F55
DOI: https://doi.org/10.1090/S0002-9939-2014-11992-7
Published electronically: June 2, 2014
MathSciNet review: 3223348
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type $ B$ and, more generally, for the complex reflection groups $ G(m,1,n)$ is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys-Murphy elements.


References [Enhancements On Off] (What's this?)

  • [1] I. V. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 87-88 (Russian). MR 831062 (87m:22031)
  • [2] I. V. Cherednik, Calculation of the monodromy of some $ W$-invariant local systems of type $ B,C$ and $ D$, Funktsional. Anal. i Prilozhen. 24 (1990), no. 1, 88-89 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 1, 78-79. MR 1052280 (91i:17019), https://doi.org/10.1007/BF01077930
  • [3] James Grime, The hook fusion procedure, Electron. J. Combin. 12 (2005), Research Paper 26, 14 pp. (electronic). MR 2156680 (2006m:05255)
  • [4] James Grime, The hook fusion procedure for Hecke algebras, J. Algebra 309 (2007), no. 2, 744-759. MR 2303204 (2008a:05270), https://doi.org/10.1016/j.jalgebra.2006.06.024
  • [5] Valentina Guizzi and Paolo Papi, A combinatorial approach to the fusion process for the symmetric group, European J. Combin. 19 (1998), no. 7, 835-845. MR 1649970 (2000g:05149), https://doi.org/10.1006/eujc.1998.0242
  • [6] A. P. Isaev and A. I. Molev, Fusion procedure for the Brauer algebra, Algebra i Analiz 22 (2010), no. 3, 142-154. English translation in St. Petersburg Math. J. 22 (2011), no. 3, 437-446. MR 2729943 (2011i:16043)
  • [7] A. P. Isaev, A. I. Molev, and A. F. Oskin, On the idempotents of Hecke algebras, Lett. Math. Phys. 85 (2008), no. 1, 79-90. MR 2425663 (2009e:20010), https://doi.org/10.1007/s11005-008-0254-7
  • [8] A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, A new fusion procedure for the Brauer algebra and evaluation homomorphisms, Int. Math. Res. Not. IMRN 2012, no. 11, 2571-2606. MR 2926990
  • [9] A. Isaev, A. Molev and O. Ogievetsky, Idempotents for Birman-Murakami-Wenzl algebras and reflection equation, arXiv:1111.2502
  • [10] M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, The $ A^{(1)}_n$ face models, Comm. Math. Phys. 119 (1988), no. 4, 543-565. MR 973016 (90h:17044)
  • [11] Andrew R. Jones, The structure of the Young symmetrizers for spin representations of the symmetric group. I, J. Algebra 205 (1998), no. 2, 626-660. MR 1632785 (99j:20017), https://doi.org/10.1006/jabr.1997.7400
  • [12] A. R. Jones and M. L. Nazarov, Affine Sergeev algebra and $ q$-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. (3) 78 (1999), no. 3, 481-512. MR 1674836 (2000a:20021), https://doi.org/10.1112/S002461159900177X
  • [13] A. A. Jucis, On the Young operators of symmetric groups, Litovsk. Fiz. Sb. 6 (1966), 163-180 (Russian, with Lithuanian and English summaries). MR 0202866 (34 #2725)
  • [14] A. I. Molev, On the fusion procedure for the symmetric group, Rep. Math. Phys. 61 (2008), no. 2, 181-188. MR 2424084 (2009f:20012), https://doi.org/10.1016/S0034-4877(08)80005-5
  • [15] Maxim Nazarov, Young's symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190-257. MR 1448714 (98m:20019), https://doi.org/10.1006/aima.1997.1621
  • [16] Maxim Nazarov, Yangians and Capelli identities, Kirillov's seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 139-163. MR 1618751 (99g:17033)
  • [17] Maxim Nazarov, Mixed hook-length formula for degenerate affine Hecke algebras, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001), Lecture Notes in Math., vol. 1815, Springer, Berlin, 2003, pp. 223-236. MR 2009842 (2004k:20012), https://doi.org/10.1007/3-540-44890-X_10
  • [18] Maxim Nazarov, A mixed hook-length formula for affine Hecke algebras, European J. Combin. 25 (2004), no. 8, 1345-1376. MR 2095485 (2005g:20010), https://doi.org/10.1016/j.ejc.2003.10.010
  • [19] Maxim Nazarov and Vitaly Tarasov, On irreducibility of tensor products of Yangian modules associated with skew Young diagrams, Duke Math. J. 112 (2002), no. 2, 343-378. MR 1894364 (2003b:17021), https://doi.org/10.1215/S0012-9074-02-11225-3
  • [20] O. V. Ogievetsky and L. Poulain d'Andecy, On representations of cyclotomic Hecke algebras, Modern Phys. Lett. A 26 (2011), no. 11, 795-803. MR 2795629 (2012g:20011), https://doi.org/10.1142/S0217732311035377
  • [21] O. V. Ogievetsky and L. Poulain d'Andecy, Cyclotomic Hecke algebras: Jucys-Murphy elements, representations, classical limit, to appear.
  • [22] I. A. Pushkarev, On the theory of representations of the wreath products of finite groups and symmetric groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 229-244, 294-295 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 96 (1999), no. 5, 3590-3599. MR 1691647 (2000c:20026), https://doi.org/10.1007/BF02175835
  • [23] Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99-133. MR 1444315 (98d:20007), https://doi.org/10.1112/S0024611597000282
  • [24] Weiqiang Wang, Vertex algebras and the class algebras of wreath products, Proc. London Math. Soc. (3) 88 (2004), no. 2, 381-404. MR 2032512 (2005a:17029), https://doi.org/10.1112/S0024611503014382
  • [25] Alfred Young, On Quantitative Substitutional Analysis, Proc. London Math. Soc. S2-31, no. 1, 273. MR 1577466, https://doi.org/10.1112/plms/s2-31.1.273

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20C05, 20F55

Retrieve articles in all journals with MSC (2010): 20C05, 20F55


Additional Information

O. V. Ogievetsky
Affiliation: Aix Marseille University, Center of Theoretical Physics, UMR 7332, Luminy, 13288 Marseille, France (On leave of absence from P. N. Lebedev Physical Institute, Leninsky Pr. 53,117924 Moscow, Russia)
Email: oleg@cpt.univ-mrs.fr

L. Poulain d’Andecy
Affiliation: Aix Marseille University, Center of Theoretical Physics, UMR 7332, Luminy, 13288 Marseille, France
Address at time of publication: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P. O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: lpoulain@cpt.univ-mrs.fr, L.B.PoulainDAndecy@uva.nl

DOI: https://doi.org/10.1090/S0002-9939-2014-11992-7
Received by editor(s): May 10, 2012
Received by editor(s) in revised form: August 10, 2012
Published electronically: June 2, 2014
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society