Skip to Main Content

Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2024 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Computation in Coxeter groups II. Constructing minimal roots
HTML articles powered by AMS MathViewer

by Bill Casselman
Represent. Theory 12 (2008), 260-293
DOI: https://doi.org/10.1090/S1088-4165-07-00319-6
Published electronically: August 19, 2008
PDF | Request permission

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
  • 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
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
Bibliographic Information
  • Bill Casselman
  • Affiliation: Mathematics Department, University of British Columbia, Vancouver, Canada
  • MR Author ID: 46050
  • Email: cass@math.ubc.ca
  • 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