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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: 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.


References [Enhancements On Off] (What's this?)

  • 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
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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.