On the Y555 complex reflection group

TL;DR

This paper provides a computer-free proof that a specific complex reflection group related to the Y555 diagram is the full isometry group of a lattice over Eisenstein integers, also classifying related lattices.

Contribution

It offers a novel, computer-free proof of the structure of the Y555 complex reflection group and classifies associated lattices over Eisenstein integers.

Findings

The group generated by 16 complex reflections of order 3 is the full isometry group of a specific lattice.

Enumeration of cusps of the lattice over Eisenstein integers.

Classification of root and Niemeier lattices over Eisenstein integers.

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.