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

DOI:
https://doi.org/10.1090/S0002-9947-2012-05517-6

Published electronically:
April 25, 2012

MathSciNet review:
2922614

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Abstract | References | Similar Articles | Additional Information

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 : there are only four such lattices, namely, the -lattices whose real forms are , , and . 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 , picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally, we observe that if is primitive and has a set of roots whose -span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for . The group 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 and , new diagrams are obtained. For , our new diagram extends to an ``affine diagram'' with symmetry.

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Additional Information

**Tathagata Basak**

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

Email:
tathastu@gmail.com

DOI:
https://doi.org/10.1090/S0002-9947-2012-05517-6

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.