On Coxeter Diagrams of complex reflection groups

TL;DR

This paper investigates Coxeter diagrams of complex reflection groups, providing a new classification proof for certain root lattices, and introduces an algorithm to characterize diagrams and generate minimal sets of reflections, revealing new diagrams and symmetries.

Contribution

It offers a combinatorial proof for classifying specific root lattices and develops an algorithm to identify diagrams and minimal generating sets for unitary reflection groups.

Findings

Classified four root lattices over 5; lattices with real forms A2, D4, E6, E8.

Developed an algorithm to extract complex reflections and simple roots from unitary reflection groups.

Discovered new diagrams for groups G_{33} and G_{34}, including an affine diagram with rac{1}{7}rac{1}{7} symmetry.

Abstract

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices, namely, the $\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou\'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an "affine diagram" with $\ZZ/7\ZZ$ symmetry.