On the Y555 complex reflection group

Abstract

We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 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[cube root of 1]. Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over this ring.