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.
- 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: http://www.maths.usyd.edu.au:8000/res/Algebra/Bri/dom-min-roots.html
- Brigitte Brink, The set of dominance-minimal roots, J. Algebra 206 (1998), no. 2, 371–412. MR 1637139, DOI 10.1006/jabr.1997.7418
- 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 10.1007/BF01445101
- 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 http://www.math.ubc.ca/~cass/coxeter.tar.gz
- 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
- Bill Casselman
- Affiliation: Mathematics Department, University of British Columbia, Vancouver, Canada
- MR Author ID: 46050
- Email: firstname.lastname@example.org
- Received by editor(s): February 20, 2005
- Received by editor(s) in revised form: August 20, 2006
- Published electronically: August 19, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Represent. Theory 12 (2008), 260-293
- MSC (2000): Primary 20F55
- DOI: https://doi.org/10.1090/S1088-4165-07-00319-6
- MathSciNet review: 2439007