Decomposition matrices for the generic Hecke algebras on 3 strands in characteristic 0. (English) Zbl 1454.20008

Summary: We determine all the decomposition matrices of the generic Hecke algebras on 3 strands in characteristic 0. These are the generic Hecke algebras associated with the exceptional complex reflection groups \(G_4, G_8\), and \(G_{16}\). We prove that for every choice of the parameters that define these algebras, all simple representations of the specialized algebra are obtained as modular reductions of simple representations of the generic algebra.


20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F36 Braid groups; Artin groups


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[1] Boura, C., Chavli, E., Chlouveraki, M., Karvounis, K.: The BMM symmetrising trace conjecture for groups G4, G5, G6, G7, G8. J Symb Comput. (2019) doi:10.1016/j.jsc.2019.02.012 · Zbl 1453.20008
[2] Broué, M.; Malle, G., Zyklotomische Heckealgebren, Astérisque, 212, 119-189 (1993) · Zbl 0835.20064
[3] Broué, M.; Malle, G.; Michel, J., Towards Spetses I, Trans. Groups, 4, 2-3, 157-218 (1999) · Zbl 0972.20024
[4] Broué, M.; Malle, G.; Rouquier, R., Complex reflection groups, braid groups, Hecke algebras, J. reine angew. Math., 500, 127-190 (1998) · Zbl 0921.20046
[5] Chavli, E., Universal deformations of the finite quotients of the braid group on 3 strands, J. Algebra, 459, 238-271 (2016) · Zbl 1345.20048
[6] Chlouveraki, M., Blocks and families for cyclotomic Hecke algebras, Lecture Notes in Mathematics, 2009-2009 (1981), Berlin: Springer, Berlin · Zbl 0461.14010
[7] Chlouveraki, M.; Miyachi, H., Decomposition matrices for d-Harish-Chandra series: the exceptional rank two cases, LMS J. Comput. Math., 14, 271-290 (2011) · Zbl 1296.20003
[8] Coxeter, H.S.M.: Factor groups of the braid group. In: Proceedings of Fourth Canad. Math. Congress, pp. 95-122 (1957)
[9] Geck, M.; Pfeiffer, G., Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs New Series 21 (2000), New York: Oxford University Press, New York · Zbl 0996.20004
[10] Malle, G., On the rationality and fake degrees of characters of cyclotomic algebras, J. Math. Sci. Univ. Tokyo, 6, 647-677 (1999) · Zbl 0964.20003
[11] Malle, G.; Michel, J., Constructing representations of Hecke algebras for complex reflection groups, LMS J. Comput. Math., 13, 426-450 (2010) · Zbl 1225.20002
[12] Malle, G.; Rouquier, R., Familles de caractères de groupes de réflexions complexes, Represent. Theory, 7, 610-640 (2003) · Zbl 1072.20007
[13] Marin, I., Wagner, E.: Markov traces on the B,irman-Wenzl-Murakami algebras, preprint, arXiv:1403.4021
[14] Meliot, P-L, Representation theory of symmetric groups (2017), Boca Raton: CRC Press, Boca Raton
[15] Michel, J., The development version of the CHEVIE package of GAP3, J. Algebra, 435, 308-336 (2015) · Zbl 1322.20002
[16] Serre, J-P, Linear representations of finite groups, vol. 42 (2012), Berlin: Springer Science & Business Media, Berlin
[17] Shephard, GC; Todd, JA, Finite unitary reflection groups, C.nad. J. Math., 6, 274-304 (1954) · Zbl 0055.14305
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