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*, J. Algebra**206**(1998), no. 2, 371–412. MR**1637139**, https://doi.org/10.1006/jabr.1997.7418**4.**Brigitte Brink and Robert B. Howlett,*A finiteness property and an automatic structure for Coxeter groups*, Math. Ann.**296**(1993), no. 1, 179–190. MR**1213378**, https://doi.org/10.1007/BF01445101**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*, Electron. J. Combin.**9**(2002), no. 1, Research Paper 25, 22. MR**1912807****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.**Jacques Tits,*Le problème des mots dans les groupes de Coxeter*, Symposia Mathematica (INDAM, Rome, 1967/68) Academic Press, London, 1969, pp. 175–185 (French). MR**0254129****10.**È. B. Vinberg,*Discrete linear groups that are generated by reflections*, Izv. Akad. Nauk SSSR Ser. Mat.**35**(1971), 1072–1112 (Russian). MR**0302779**

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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.