Computation in Coxeter groups II. Constructing minimal roots

Author:
Bill Casselman

Journal:
Represent. Theory **12** (2008), 260-293

MSC (2000):
Primary 20F55

DOI:
https://doi.org/10.1090/S1088-4165-07-00319-6

Published electronically:
August 19, 2008

MathSciNet review:
2439007

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Abstract | References | Similar Articles | Additional Information

Abstract: In an earlier paper (Casselman, 2002) I described how a number of ideas due to Fokko du Cloux and myself could be incorporated into a reasonably efficient program to carry out multiplication in arbitrary Coxeter groups. At the end of that paper I discussed how this algorithm could be used to build the reflection table of minimal roots, which could in turn form the basis of a much more efficient multiplication algorithm. In this paper, following a suggestion of Robert Howlett, I explain how results due to Brigitte Brink can be used to construct the minimal root reflection table directly and more efficiently.

**1.**Brigitte Brink, `On root systems and automaticity of Coxeter groups', Ph.D. thesis, University of Sydney, 1994.**2.**Brigitte Brink, `The set of dominance-minimal roots', available as Report 94-43 from the School of Mathematics and Statistics at the University of Sydney:`http://www.maths. usyd.edu.au:8000/res/Algebra/Bri/dom-min-roots.html`**3.**Brigitte Brink, `The set of dominance-minimal roots',*Journal of Algebra***206**(1998), 371-412. This publication, although it has the same title as the previous item, is very different from it. For purposes of computation it is decidedly less interesting, since it doesn't explicitly introduce the indecomposable minimal roots and emphasizes hand-lists instead. MR**1637139 (99k:20083)****4.**Brigitte Brink and Robert Howlett, `A finiteness property and an automatic structure for Coxeter groups',*Math. Ann.***296**(1993), 179-190. MR**1213378 (94d:20045)****5.**Bill Casselman, `Automata to perform basic calculations in Coxeter groups', in*Representations of Groups*,*CMS Conference Proceedings 16*, Amer. Math. Soc., Providence, RI, 1994.**6.**Bill Casselman, `Computation in Coxeter groups. I. Multiplication'.*Electronic Journal of Combinatorics***9(1)**(2002), #25. MR**1912807 (2003h:20072)****7.**Bill Casselman, `Java code for finding minimal roots', at`http://www.math.ubc.ca/ ~cass/coxeter.tar.gz`**8.**Fokko du Cloux, `Un algorithme de forme normale pour les groupes de Coxeter', preprint, Centre de Mathématiques à l'École Polytechnique, 1990.**9.**J. Tits, `Le problème des mots dans les groupes de Coxeter',*Symposia Math.***1**(1968), 175-185. MR**0254129 (40:7339)****10.**E. B. Vinberg, `Discrete linear groups generated by reflections',*Math. USSR Izvestia***5**(1971), 1083-1119. MR**0302779 (46:1922)**

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Additional Information

**Bill Casselman**

Affiliation:
Mathematics Department, University of British Columbia, Vancouver, Canada

Email:
cass@math.ubc.ca

DOI:
https://doi.org/10.1090/S1088-4165-07-00319-6

Received by editor(s):
February 20, 2005

Received by editor(s) in revised form:
August 20, 2006

Published electronically:
August 19, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.