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Summary: ON THE Y555 COMPLEX REFLECTION GROUP
DANIEL ALLCOCK
Abstract. We give a computer-free proof of a theorem of Basak,
describing the group generated by 16 complex reflections of or-
der 3, satisfying the braid and commutation relations of the Y555
diagram. The group is the full isometry group of a certain lattice
of signature (13, 1) over the Eisenstein integers Z[ 3
1]. Along the
way we enumerate the cusps of this lattice and classify the root
and Niemeier lattices over Z[ 3
1]
The author has conjectured [3] that the largest sporadic finite simple
group, the monster, is related to complex algebraic geometry, with a
certain complex hyperbolic orbifold acting as a sort of intermediary.
Specifically, the bimonster (M × M):2 and a certain group P acting
on complex hyperbolic 13-space are conjecturally both quotients of
1 (CH13
-)/P for a certain hyperplane arrangement in CH13
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