Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Computation in Coxeter groups II. Constructing minimal roots

Author: Bill Casselman
Journal: Represent. Theory 12 (2008), 260-293
MSC (2000): Primary 20F55
Published electronically: August 19, 2008
MathSciNet review: 2439007
Full-text PDF Free Access

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.

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

  • Brigitte Brink, ‘On root systems and automaticity of Coxeter groups’, \frenchspacing Ph.D. thesis, University of Sydney, 1994.
  • Brigitte Brink, ‘The set of dominance-minimal roots’, available as Report 94–43 from theSchool of Mathematics and Statistics at the University of Sydney:
  • Brigitte Brink, The set of dominance-minimal roots, J. Algebra 206 (1998), no. 2, 371–412. MR 1637139, DOI
  • 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, DOI
  • Bill Casselman, ‘Automata to perform basic calculations in Coxeter groups’, in Representations of Groups, CMS Conference Proceedings 16, Amer. Math. Soc., Providence, RI, 1994.
  • Bill Casselman, Computation in Coxeter groups. I. Multiplication, Electron. J. Combin. 9 (2002), no. 1, Research Paper 25, 22. MR 1912807
  • Bill Casselman, ‘Java code for finding minimal roots’, at
  • Fokko du Cloux, ‘Un algorithme de forme normale pour les groupes de Coxeter’, preprint, Centre de Mathématiques à l’École Polytechnique, 1990.
  • 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
  • È. B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1072–1112 (Russian). MR 0302779

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20F55

Retrieve articles in all journals with MSC (2000): 20F55

Additional Information

Bill Casselman
Affiliation: Mathematics Department, University of British Columbia, Vancouver, Canada
MR Author ID: 46050

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.