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The BMR freeness conjecture for the 2-reflection groups. (English) Zbl 1419.20003
Summary: We prove the freeness conjecture of Broué, Malle and Rouquier [M. Broué et al., J. Reine Angew. Math. 500, 127–190 (1998; Zbl 0921.20046)] for the Hecke algebras associated to the primitive complex 2-reflection groups with a single conjugacy class of reflections.

##### MSC:
 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects)
CHEVIE
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##### References:
 [1] Bannai, Etsuko, Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension $$2$$, J. Math. Soc. Japan, 28, 3, 447-454, (1976) · Zbl 0326.57015 [2] Bessis, David, Finite complex reflection arrangements are $$K(π ,1)$$, Ann. of Math. (2), 181, 3, 809-904, (2015) · Zbl 1372.20036 [3] Bessis, David; Michel, Jean, Explicit presentations for exceptional braid groups, Experiment. Math., 13, 3, 257-266, (2004) · Zbl 1092.20033 [4] Brou\'e, Michel; Malle, Gunter; Rouquier, Rapha\"el, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math., 500, 127-190, (1998) · Zbl 0921.20046 [5] [C]CHAVLI E. Chavli, The Brou\'e-Malle-Rouquier conjecture for reflection groups of rank $$2$$, Th\ese de doctorat, Universit\'e Paris-Diderot, 2016, arXiv:1608.00834. · Zbl 1345.20048 [6] Digne, Fran\ccois; Marin, Ivan; Michel, Jean, The center of pure complex braid groups, J. Algebra, 347, 206-213, (2011) · Zbl 1241.20039 [7] Etingof, Pavel; Rains, Eric, Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups, J. Algebra, 299, 2, 570-588, (2006) · Zbl 1171.16007 [8] Geck, Meinolf; Pfeiffer, G\"otz, Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs. New Series 21, xvi+446 pp., (2000), The Clarendon Press, Oxford University Press, New York · Zbl 0996.20004 [9] Losev, Ivan, Finite-dimensional quotients of Hecke algebras, Algebra Number Theory, 9, 2, 493-502, (2015) · Zbl 1323.20008 [10] Marin, Ivan, The cubic Hecke algebra on at most $$5$$ strands, J. Pure Appl. Algebra, 216, 12, 2754-2782, (2012) · Zbl 1266.20006 [11] Marin, Ivan, The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group $$G_{26}$$, J. Pure Appl. Algebra, 218, 4, 704-720, (2014) · Zbl 1291.20042 [12] Michel, Jean, The development version of the CHEVIE package of GAP3, J. Algebra, 435, 308-336, (2015) · Zbl 1322.20002 [13] [P]PICTHESE M. Picantin, Petits Groupes Gaussiens, Th\ese de doctorat, Universit\'e de Caen, 2000.
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