On Coxeter diagrams of complex reflection groups
Author:
Tathagata Basak
Journal:
Trans. Amer. Math. Soc. 364 (2012), 49094936
MSC (2010):
Primary 20F55, 20F05, 20F65, 51F25
Published electronically:
April 25, 2012
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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 : 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 (Coxeterlike) 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:
http://dx.doi.org/10.1090/S000299472012055176
PII:
S 00029947(2012)055176
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.
