 
Summary: ON THE Y555 COMPLEX REFLECTION GROUP
DANIEL ALLCOCK
Abstract. We give a computerfree 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 13space are conjecturally both quotients of
1 (CH13
)/P for a certain hyperplane arrangement in CH13
