Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T09:18:47.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing in Unipotent and Reductive Algebraic Groups

Published online by Cambridge University Press:  01 February 2010

Arjeh M. Cohen ,
Sergei Haller  and
Scott H. Murray
Arjeh M. Cohen
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, Netherlands, [email protected], http://www.win.tue.nl/~amc/
Sergei Haller
Affiliation:
btexx business technologies GmbH, Rheinstraβe 4N, 55116 Mainz, Germany, [email protected], http://www.sergei-haller.de
Scott H. Murray
Affiliation:
School of Mathematics and Statistics F07, Faculty of Science, University of Sydney, NSW 2006, Australia, [email protected], http://www.maths.usyd.edu.au/u/murray/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 343 - 366
Copyright
Copyright © London Mathematical Society 2008

References

1.Bosma, Wieb, Cannon, John and Playoust, Catherine, ‘The Magma algebra system. I. The user language’, Computational algebra and number theory, London, 1993, J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
2.Carter, Roger W., Simple groups of Lie type, Pure and Applied Mathematics 28 (John Wiley & Sons, London-New York-Sydney, 1972).Google Scholar
3.Cohen, Arjeh M., Murray, Scott H. and Taylor, D. E., ‘Computing in groups of Lie type’, Math. Comp. 73 (2004) 1477–1498.CrossRefGoogle Scholar
4.Demazure, Michel and Gabriel, Pierre, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs (Masson & Cie, Paris, 1970). Avec un appendice Corps de classes local par Michiel Hazewinkel.Google Scholar
5.Hall, Philip, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes (Mathematics Department, Queen Mary College, London, 1969).Google Scholar
6.Hall, Marshall Jr.., The theory of groups (Chelsea, New York, 1976). Reprinting of the 1968 edition.Google Scholar
7.Holt, Derek F., Eick, Bettina and O'Brien, Eamonn A., Handbook of computational group theory. Discrete Mathematics and its Applications, Boca Raton (Chapman & Hall/CRC, Boca Raton, FL, 2005). ISBN 1-58488-372-3.CrossRefGoogle Scholar
8.Humphreys, James E., Introduction to Lie algebras and representation theory (Springer, New York, 1978). ISBN 0-387-90053-5, 2nd printing, revised.Google Scholar
9.Kernighan, Brian W. and Ritchie, Dennis M., The C programming language, 2nd edn, Prentice-Hall Internat. Ser. Comput. Sci. (Prentice-Hall, Englewood Cliffs, NJ, 1988).Google Scholar
10.Leedham-Green, C. R. and Soicher, L. H., ‘Collection from the left and other strategies’, J. Symbolic Comput. 9 (1990) 665–675. Computational group theory, Part 1.CrossRefGoogle Scholar
11.Leedham-Green, C. R. and Soicher, Leonard H., ‘Symbolic collection using Deep Thought’, LMS J. Comput. Math. 1 (1998) 9–24 (electronic).CrossRefGoogle Scholar
12.Leedham-Gren, Charles R., ‘The computational matrix group project’, Groups and computation III, Columbus, OH, 1999, Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, Berlin, 2001) 229–247.CrossRefGoogle Scholar
13.Merkwitz, Wolfgang Wilhelm, ‘Symbolische Multiplikation in nilpotenten Gruppen mit Deep Thought’, Master's thesis, Rheinisch-Westfälische Technische Hochschule Aachen (1997).Google Scholar
14.Papi, Paolo, ‘A characterization of a special ordering in a root system’, Proc. Amer. Math. Soc. 120 (1994) 661–665.CrossRefGoogle Scholar
15.Rosenlicht, Maxwell, ‘Some rationality questions on algebraic groups’. Ann. Mat. Pura Appl. (4) 43 (1957) 25–50.CrossRefGoogle Scholar
16.Serre, Jean-Pierre, Algebraic groups and class fields, Graduate Texts in Mathematics 117 (Springer-Verlag, New York, 1988). ISBN 0-387-96648-X, translated from the French.CrossRefGoogle Scholar
17.Springer, T. A., Linear algebraic groups, 2nd edn (Birkhäuser, Boston, MA, 1998). ISBN 0-8176-4021-5.CrossRefGoogle Scholar
18.Steinberg, R., ‘Lectures on Chevalley groups’, Technical Report, Yale University (1968).Google Scholar
19.Vaughan-Lee, M. R., ‘Collection from the left’ J. Symbolic Comput. 9 (1990) 725–733. Computational group theory, Part 1.CrossRefGoogle Scholar
20.Waterhouse, William C., Introduction to affine group schemes, Graduate Texts in Mathematics 66 (Springer, New York, 1979). ISBN 0-387-90421-2.CrossRefGoogle Scholar