Geodesic
An inductive approach to representations of complex reflection groups
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 109-124
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We propose an inductive approach to the representation theory of the chain of complex reflection groups $G(m,1,n)$. We obtain the Jucys–Murphy elements of $G(m,1,n)$ from the Jucys–Murphy elements of the cyclotomic Hecke algebra and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. We construct representations of $G(m,1,n)$ using a new associative algebra whose underlying vector space is the tensor product of the group ring $\mathbb{C}G(m,1,n)$ with a free associative algebra generated by the standard $m$-tableaux.
Keywords: group tower, Hecke algebra, reflection group, Young diagram
Mots-clés : maximal commutative subalgebra, Young tableau. 
@article{TMF_2013_174_1_a7,
     author = {O. V. Ogievetskii and L. Poulain d'Andecy},
     title = {An~inductive approach to representations of complex reflection groups},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {109--124},
     year = {2013},
     volume = {174},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a7/}
}
TY  - JOUR
AU  - O. V. Ogievetskii
AU  - L. Poulain d'Andecy
TI  - An inductive approach to representations of complex reflection groups
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2013
SP  - 109
EP  - 124
VL  - 174
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a7/
LA  - ru
ID  - TMF_2013_174_1_a7
ER  - 
%0 Journal Article
%A O. V. Ogievetskii
%A L. Poulain d'Andecy
%T An inductive approach to representations of complex reflection groups
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2013
%P 109-124
%V 174
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a7/
%G ru
%F TMF_2013_174_1_a7
O. V. Ogievetskii; L. Poulain d'Andecy. An inductive approach to representations of complex reflection groups. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 109-124. http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a7/

[1] G. C. Shephard, J. A. Todd, Canadian J. Math., 6 (1954), 274–304 | DOI | MR | Zbl

[2] S. Ariki, K. Koike, Adv. Math., 106:2 (1994), 216–243 | DOI | MR | Zbl

[3] M. Broué, G. Malle, Astérisque, 212 (1993), 119–189 | MR | Zbl

[4] I. V. Cherednik, Duke Math. J., 54:2 (1987), 563–577 | DOI | MR | Zbl

[5] P. Hoefsmit, Representations of Hecke algebras of finite groups with BN-pairs of classical type, Ph. D. Thesis, University of British Columbia, Vancouver, 1974 | MR

[6] A. Okounkov, A. Vershik, Selecta Math. (N. S.), 2:4 (1996), 581–605 | DOI | MR | Zbl

[7] O. V. Ogievetsky, L. Poulain d'Andecy, Modern Phys. Lett. A, 26:11 (2011), 795–803, arXiv: 1012.5844 | DOI | MR | Zbl

[8] W. Specht, Schriften Math. Seminar (Berlin), 1 (1932), 1–32 | Zbl

[9] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley, Reading, MA, 1981 | MR | Zbl

[10] I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1998 | MR | Zbl

[11] I. A. Pushkarev, “K teorii predstavlenii spletenii konechnykh grupp s simmetricheskimi gruppami”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. II, Zap. nauchn. sem. POMI, 240, POMI, SPb., 1997, 229–244 | MR | Zbl

[12] A. Young, Proc. London Math. Soc. (2), 31:1 (1930), 273–288 | DOI | MR

[13] O. V. Ogievetsky, L. Poulain d'Andecy, Jucys–Murphy elements and representations of cyclotomic Hecke algebras, arXiv: 1206.0612 | MR

[14] A. Ram, Proc. London Math. Soc., 75:1 (1997), 99–133, arXiv: math.RT/9511223 | DOI | MR | Zbl

[15] W. Wang, Proc. London Math. Soc., 88:2 (2004), 381–404, arXiv: math.QA/0203004 | DOI | MR | Zbl

[16] J. Wan, W. Wang, Int. Math. Res. Not., 2008, 128, 31 pp., arXiv: 0806.0196 | MR

[17] A. Ram, A. Shepler, Comment. Math. Helv., 78:2 (2003), 308–334, arXiv: math.GR/0209135 | DOI | MR | Zbl

[18] A. P. Isaev, O. V. Ogievetsky, Czechoslovak J. Phys., 55:11 (2005), 1433–1441 | DOI | MR