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On the \(Y_{555}\) complex reflection group. (English) Zbl 1269.20029

Summary: We give a computer-free proof of a theorem of T. Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the \(Y_{555}\) diagram, [J. Algebra 309, No. 1, 32-56 (2007; Zbl 1125.11040)]. The group is the full isometry group of a certain lattice of signature \((13,1)\) over the Eisenstein integers \(\mathbb Z(\root 3\of 1)\). Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over \(\mathbb Z(\root 3\of 1)\).

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
11H56 Automorphism groups of lattices
11H06 Lattices and convex bodies (number-theoretic aspects)
20D08 Simple groups: sporadic groups

Citations:

Zbl 1125.11040
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References:

[1] Allcock, D., The Leech lattice and complex hyperbolic reflections, Invent. Math., 140, 283-301 (2000) · Zbl 1012.11053
[2] D. Allcock, A monstrous proposal, in: J. Harnad (Ed.), Groups and Symmetries: From the Neolithic Scots to John McKay, Amer. Math. Soc., in press, available at arXiv:math.GR/0606043; D. Allcock, A monstrous proposal, in: J. Harnad (Ed.), Groups and Symmetries: From the Neolithic Scots to John McKay, Amer. Math. Soc., in press, available at arXiv:math.GR/0606043 · Zbl 1193.20015
[3] Basak, T., The complex Lorentzian Leech lattice and the bimonster, J. Algebra, 309, 32-56 (2007) · Zbl 1125.11040
[4] T. Basak, The complex Lorentzian Leech lattice and the bimonster (II), preprint, 2008, arXiv:math.GR/0811.0062; T. Basak, The complex Lorentzian Leech lattice and the bimonster (II), preprint, 2008, arXiv:math.GR/0811.0062 · Zbl 1345.20055
[5] Cohen, A., Finite complex reflection groups, Ann. Sci. École Norm. Sup.. Ann. Sci. École Norm. Sup., Ann. Sci. École Norm. Sup., 11, 4, 613-436 (1978), (Erratum) · Zbl 0396.20033
[6] Conway, J. H., A characterization of Leech’s lattice, Invent. Math., 7, 137-142 (1969), also as Chapter 12 of [10] · Zbl 0212.07001
[7] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., ATLAS of Finite Groups (1985), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0568.20001
[8] Conway, J. H.; Norton, S. P.; Soicher, L. H., The bimonster, the group \(Y_{555}\) and the projective plane of order 3, (Tangora, M. C., Computers in Algebra (1988), Dekker: Dekker New York), 27-50 · Zbl 0693.20014
[9] Conway, J. H.; Pritchard, A. D., Hyperbolic reflexions for the bimonster and \(3 Fi_{24}\), (Groups, Combinatoric. Groups, Combinatoric, Durham, 1990. Groups, Combinatoric. Groups, Combinatoric, Durham, 1990, London Math. Soc. Lecture Note Ser., vol. 165 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 24-45 · Zbl 0836.20013
[10] Conway, J. H.; Sloane, N. J.A., Sphere Packings, Lattices and Groups (1993), Springer · Zbl 0785.11036
[11] Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. Inst. Hautes Ètudes Sci., 63, 5-89 (1986) · Zbl 0615.22008
[12] Ivanov, A. A., A geometric characterization of the Monster, (Groups, Combinatorics & Geometry. Groups, Combinatorics & Geometry, Durham, 1990. Groups, Combinatorics & Geometry. Groups, Combinatorics & Geometry, Durham, 1990, London Math. Soc. Lecture Note Ser., vol. 165 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 46-62 · Zbl 0821.20005
[13] Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, Publ. Math. Inst. Hautes Ètudes Sci., 63, 91-106 (1986) · Zbl 0615.22009
[14] Nikulin, V. V., On the classification of hyperbolic root systems of rank three, Proc. Steklov Inst. Math., 230, 3, 1-241 (2000) · Zbl 0997.17014
[15] Norton, S. P., Constructing the Monster, (Groups, Combinatorics & Geometry. Groups, Combinatorics & Geometry, Durham, 1990. Groups, Combinatorics & Geometry. Groups, Combinatorics & Geometry, Durham, 1990, London Math. Soc. Lecture Note Ser., vol. 165 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 63-76 · Zbl 0806.20019
[16] Shephard, G. C.; Todd, J. A., Finite unitary reflection groups, Canad. J. Math., 6, 274-304 (1954) · Zbl 0055.14305
[17] Venkov, B. B., On the classification of integral even unimodular 24-dimensional quadratic forms, Proc. Steklov Inst. Math., 4, 63-74 (1980), also as Chapter 18 of [10] · Zbl 0443.10021
[18] Vinberg, E. B., The unimodular integral quadratic forms, Funkcional. Anal. i Priložen., 6, 2, 24-31 (1972), (in Russian) · Zbl 0252.10027
[19] Vinberg, E. B., The groups of units of certain quadratic forms, Mat. USSR Sb., 87, 17-35 (1972) · Zbl 0252.20054
[20] Wilson, R. A., The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra, 77, 449-462 (1982)
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