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On Coxeter diagrams of complex reflection groups

Author: Tathagata Basak
Journal: Trans. Amer. Math. Soc. 364 (2012), 4909-4936
MSC (2010): Primary 20F55, 20F05, 20F65, 51F25
Published electronically: April 25, 2012
MathSciNet review: 2922614
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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 $ \mathcal {E} = \mathbb{Z}[e^{ 2 \pi i/3}]$: there are only four such lattices, namely, the $ \mathcal {E}$-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 Broué, 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 $ \mathbb{Z}$-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 $ \mathbb{Z}/7\mathbb{Z}$ symmetry.

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Additional Information

Tathagata Basak
Affiliation: Department of Mathematics, Iowa State University, Carver Hall, Ames, Iowa 50011

Keywords: Unitary reflection group, Coxeter diagram, Weyl group, simple root
Received by editor(s): September 25, 2009
Received by editor(s) in revised form: July 27, 2010, and December 3, 2010
Published electronically: April 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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