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Decomposition matrices for d-Harish-Chandra series: the exceptional rank two cases

Part of: Representation theory of groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  01 November 2011

Maria Chlouveraki  and
Hyohe Miyachi
Maria Chlouveraki
Affiliation:
School of Mathematics, University of Edinburgh, JCMB, Room 5610 King’s Buildings Edinburgh EH9 3JZ, United Kingdom (email: [email protected])
Hyohe Miyachi
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya Aichi, 464-8602, Japan (email: [email protected])

Abstract

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We calculate all decomposition matrices of the cyclotomic Hecke algebras of the rank two exceptional complex reflection groups in characteristic zero. We prove the existence of canonical basic sets in the sense of Geck–Rouquier and show that all modular irreducible representations can be lifted to the ordinary ones.

MSC classification

Secondary: 20C08: Hecke algebras and their representations 20G40: Linear algebraic groups over finite fields 20C20: Modular representations and characters
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 271 - 290
Copyright
Copyright © London Mathematical Society 2011

References

[1]Ariki, S., ‘On the decomposition numbers of the Hecke algebra of G(m,1,n)’, J. Math. Kyoto Univ. 36 (1996) 789–808.Google Scholar
[2]Ariki, S. and Mathas, A., ‘The representation type of Hecke algebras of type B’, Adv. Math. 181 (2004) 134–159.CrossRefGoogle Scholar
[3]Broué, M., ‘Equivalences of blocks of group algebras’, Finite-dimensional algebras and related topics (Ottawa, ON, 1992), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 424 (Kluwer Academic, Dordrecht, 1994) 1–26.Google Scholar
[4]Broué, M. and Malle, G., ‘Zyklotomische Heckealgebren’, Astérisque 212 (1993) 119–189.Google Scholar
[5]Broué, M., Malle, G. and Michel, J., ‘Generic blocks of finite reductive groups’, Astérisque 212 (1993) 7–92.Google Scholar
[6]Broué, M., Malle, G. and Michel, J., ‘Towards spetses I’, Transform. Groups 4 (1999) no. 2–3, 157–218.CrossRefGoogle Scholar
[7]Broué, M., Malle, G. and Rouquier, R., ‘Complex reflection groups, braid groups, Hecke algebras’, J. Reine Angew. Math. 500 (1998) 127–190.Google Scholar
[8]Broué, M. and Michel, J., ‘Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne–Lusztig associées’, Finite reductive groups (Luminy, 1994), Progress in Mathematics 141 (Birkhäuser, Boston, MA, 1997) 73–139.Google Scholar
[9]Chlouveraki, M., Blocks and families for the cyclotomic Hecke algebras, Lecture Notes in Mathematics 1981 (Springer, Berlin, Heidelberg, 2009).CrossRefGoogle Scholar
[10]Chlouveraki, M. and Jacon, N., ‘Schur elements and basic sets for cyclotomic Hecke algebras’, J. Algebra Appl. 10 (2011) 979–993.CrossRefGoogle Scholar
[11]Chuang, J. and Miyachi, H., ‘Generic degree ratios at roots of unity’, in The 10th Workshop on Representation Theory of Algebraic Groups and Quantum Groups – in honor of Prof. Shoji and Prof. Shinoda’s 60th birthdays, S. Ariki et al., Preprint,http://www.math.nagoya-u.ac.jp/∼miyachi/articles/raq10pro07.pdf.Google Scholar
[12]Dipper, R., James, G. and Murphy, E., ‘Hecke algebras of type B n at roots of unity’, London Math. Soc. 70 (1995) 505–528.CrossRefGoogle Scholar
[13]Geck, M., ‘Brauer trees of Hecke algebras’, Comm. Algebra 20 (1992) 2937–2973.CrossRefGoogle Scholar
[14]Geck, M., Beiträge zur Darstellungstheorie von Iwahori–Hecke–Algebren (Habilitations-schrift, RWTH Aachen, 1993).Google Scholar
[15]Geck, M., Modular representations of Hecke algebras, Group Representation Theory (EPFL Press, Lausanne, 2007) 301–353.Google Scholar
[16]Geck, M. and Jacon, N., ‘Canonical basic sets in type B n’, J. Algebra 306 (2006) no. 1, 104–127.CrossRefGoogle Scholar
[17]Geck, M. and Pfeiffer, G., Characters of Coxeter groups and Iwahori–Hecke algebras, LMS Monographs New Series 21 (Oxford University Press, 2000).CrossRefGoogle Scholar
[18]Geck, M. and Rouquier, R., Centers and simple modules for Iwahori–Hecke algebras, Progress in Mathematics 141 (Birkhäuser, 1997) 251–272.Google Scholar
[19]Geck, M. and Rouquier, R., ‘Filtrations on projective modules for Iwahori–Hecke algebras’, Modular representation theory of finite groups (Charlottesville, VA, 1998) (de Gruyter, Berlin, 2001) 211–221.Google Scholar
[20]Genet, G. and Jacon, N., ‘Modular representations of cyclotomic Hecke algebras of type G(r,p,n)’, Int. Math. Res. Not. 2006 (2006) , doi:10.1155/IMRN/2006/93049.Google Scholar
[21]Ginzburg, V., Guay, N., Opdam, E. and Rouquier, R., ‘On the category 𝒪 for rational Cherednik algebras’, Invent. Math. 154 (2003) no. 3, 617–651.CrossRefGoogle Scholar
[22]Jacon, N., ‘On the parametrization of the simple modules for Ariki–Koike algebras at roots of unity’, J. Math. Kyoto Univ. 44 (2004) no. 4, 729–767.Google Scholar
[23]Malle, G., Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Progress in Mathematics 141 (Birkhäuser, 1996) 311–332.Google Scholar
[24]Malle, G. and Michel, J., ‘Constructing representations of Hecke algebras for complex reflection groups’, LMS J. Comput. Math. 13 (2010) 426–450.CrossRefGoogle Scholar
[25]Malle, G. and Rouquier, R., ‘Familles de caractères des groupes de réflexions complexes’, Represent. Theory 7 (2003) 610–640 (electronic).CrossRefGoogle Scholar
[26]Nagao, H. and Tsushima, Y., Representations of finite groups (Academic Press, Boston, MA, 1989).Google Scholar
[27]Uglov, D., ‘Canonical bases of higher-level q-deformed Fock spaces and Kazhdan–Lusztig polynomials’, Physical combinatorics (Kyoto, 1999), Progress in Mathematics 191 (Birkhäuser, Boston, MA, 2000) 249–299.CrossRefGoogle Scholar
[28]Uno, K., ‘On representations of nonsemisimple specialized Hecke algebras’, J. Algebra 149 (1992) no. 2, 287–312.CrossRefGoogle Scholar