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The Calogero-Moser partition for G(m, d, n)

Published online by Cambridge University Press:  11 January 2016

Gwyn Bellamy
Gwyn Bellamy*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, [email protected]
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Abstract

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We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d, n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

Keywords

16G9905E10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 207 , September 2012 , pp. 47 - 77
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[B] Bellamy, G., On singular Calogero-Moser spaces, Bull. Lond. Math. Soc. 41, no. 2 (2009), 315–326.Google Scholar
[BBR] Bessis, D., Bonnafé, C., and Rouquier, R., Quotients et extensions de groupes de r éflexion, Math. Ann. 323 (2002), 405–436.Google Scholar
[BK] Broué, M. and Kim, S., Familles de caractères des algèbres de Hecke cyclotomiques, Adv. Math. 172 (2002), 53–136.Google Scholar
[BMR] Broué, M., Malle, G., and Rouquier, R., Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190.Google Scholar
[BG] Brown, K. A. and Gordon, I. G., The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (2001), 733–779.Google Scholar
[BGS] Brown, K. A., Gordon, I. G., and Stroppel, C. H., Cherednik, Hecke and quantum algebras as free Frobenius and Calabi-Yau extensions, J. Algebra 319 (2008), 1007–1034.Google Scholar
[CH1] Chlouveraki, M., Rouquier blocks of the cyclotomic Ariki-Koike Hecke algebras, Algebra Number Theory 2 (2008), 689–720.CrossRefGoogle Scholar
[CH2] Chlouveraki, M., Blocks and Families for Cyclotomic Hecke Algebras, Lecture Notes in Math. 1981, Springer, Berlin, 2009.Google Scholar
[CH3] Chlouveraki, M., Rouquier blocks of the cyclotomic Hecke algebras of G(de,e,r), Nagoya Math J. 197 (2010), 175–212.Google Scholar
[Co] Cohen, A. M., Finite complex reflection groups, Ann. Sci. Ec. Norm. Supér. (4) 9 (1976), 379–436.CrossRefGoogle Scholar
[CR] Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Pure and Appl. Math. XI, Interscience, New York, 1962.Google Scholar
[EG] Etingof, P. and Ginzburg, V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.Google Scholar
[GP] Geck, M. and Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Math. Soc. Monogr. (N. S.) 21, Oxford University Press, New York, 2000.Google Scholar
[G1] Gordon, I. G., Baby Verma modules for rational Cherednik algebras, Bull. Lond. Math. Soc. 35 (2003), 321–336.CrossRefGoogle Scholar
[G2] Gordon, I. G., Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras, Int. Math. Res. Pap. IMRP 2008, no. 3. art. ID rpn006.Google Scholar
[GM] Gordon, I. G. and Martino, M., Calogero-Moser space, restricted rational Cherednik algebras and two-sided cells, Math. Res. Lett. 16 (2009), 255–262.Google Scholar
[JK] James, G. D. and Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, MA, 1981.Google Scholar
[Ki] Kim, S., Families of the characters of the cyclotomic Hecke algebras of G(ed, d, r), J. Algebra 289 (2005), 346–364.Google Scholar
[M] Martino, M., The Calogero-Moser partition and Rouquier families for complex reflection groups, J. Algebra 323 (2010), 193–205.Google Scholar
[Re] Read, E. W., On the finite imprimitive unitary reflection groups, J. Algebra, 45 (1977), 439–452.CrossRefGoogle Scholar
[Rot] Rotman, J. J., An Introduction to Homological Algebra, 2nd ed., Springer, New York, 2009.Google Scholar
[ST] Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.CrossRefGoogle Scholar
[Sp] Specht, W., Eine Verallgmeinerung der symmetrischen Gruppe, Sch. Math. Semin. (Berlin) 1 (1932), 1–32.Google Scholar
[Ste] Stembridge, J. R., On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140, (1989), 353–396.Google Scholar