Computing in groups of Lie type

Cohen, Arjeh; Murray, Scott; Taylor, D.

doi:10.1090/S0025-5718-03-01582-5

Computing in groups of Lie type
HTML articles powered by AMS MathViewer

by Arjeh M. Cohen, Scott H. Murray and D. E. Taylor PDF

Math. Comp. 73 (2004), 1477-1498 Request permission

Abstract:

We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

References

J. L. Alperin and Rowen B. Bell, Groups and representations, Graduate Texts in Mathematics, vol. 162, Springer-Verlag, New York, 1995. MR 1369573, DOI 10.1007/978-1-4612-0799-3
W. W. Bosma and J. J. Cannon, Handbook of magma functions, School of Mathematics and Statistics, University of Sydney, Sydney, 1997.
L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks, and P. P. Pálfy, Short presentations for finite groups, J. Algebra 194 (1997), no. 1, 79–112. MR 1461483, DOI 10.1006/jabr.1996.6980
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
G. Butler, Fundamental algorithms for permutation groups, Lecture Notes in Computer Science, vol. 559, Springer-Verlag, Berlin, 1991. MR 1225579, DOI 10.1007/3-540-54955-2
Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
M. Artin, J. E. Bertin, M. Demazure, A. Grothendieck, P. Gabriel, M. Raynaud, and J.-P. Serre, Schémas en groupes. Fasc. 6: Exposés 19 à 22, Institut des Hautes Études Scientifiques, Paris, 1965 (French). Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques, 1964, dirigé par Michel Demazure et Alexander Grothendieck. MR 0207709
Willem A. de Graaf, Lie algebras: theory and algorithms, North-Holland Mathematical Library, vol. 56, North-Holland Publishing Co., Amsterdam, 2000. MR 1743970, DOI 10.1016/S0924-6509(00)80040-9
W. A. de Graaf, Constructing representations of split semisimple Lie algebras, J. Pure Appl. Algebra 164 (2001), no. 1-2, 87–107. Effective methods in algebraic geometry (Bath, 2000). MR 1854331, DOI 10.1016/S0022-4049(00)00150-X
P. Flajolet, X. Gourdon, and D. Panario, The complete analysis of a polynomial factorization algorithm over finite fields, J. Algorithms 40 (2001), no. 1, 37–81. MR 1841252, DOI 10.1006/jagm.2001.1158
I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23–66. MR 595581, DOI 10.1007/BF01391662
The GAP Group, Aachen, St Andrews, GAP – Groups, Algorithms, and Programming, Version 4.1, 1999 (http://www.gap-system.org).
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
Peter B. Gilkey and Gary M. Seitz, Some representations of exceptional Lie algebras, Geom. Dedicata 25 (1988), no. 1-3, 407–416. Geometries and groups (Noordwijkerhout, 1986). MR 925845, DOI 10.1007/BF00191935
Sergei Haller, Unipot—a system for computing with elements of unipotent subgroups of Chevalley groups, version 1.1, Tech. report, Justus-Liebig Universität, Germany, July 2000, http://www.gap-system.org/Info4/deposit.html#pkg.
R. B. Howlett, L. J. Rylands, and D. E. Taylor, Matrix generators for exceptional groups of Lie type, J. Symbolic Comput. 31 (2001), no. 4, 429–445. MR 1823074, DOI 10.1006/jsco.2000.0431
Alexander Hulpke and Ákos Seress, Short presentations for three-dimensional unitary groups, J. Algebra 245 (2001), no. 2, 719–729. MR 1863898, DOI 10.1006/jabr.2001.8943
William M. Kantor and Ákos Seress, Black box classical groups, Mem. Amer. Math. Soc. 149 (2001), no. 708, viii+168. MR 1804385, DOI 10.1090/memo/0708
Charles R. Leedham-Green, The computational matrix group project, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 229–247. MR 1829483
C. R. Leedham-Green and L. H. Soicher, Collection from the left and other strategies, J. Symbolic Comput. 9 (1990), no. 5-6, 665–675. Computational group theory, Part 1. MR 1075430, DOI 10.1016/S0747-7171(08)80081-8
Martin W. Liebeck and Jan Saxl, On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 55 (1987), no. 2, 299–330. MR 896223, DOI 10.1093/plms/s3-55_{2}.299
R. J. Riebeek, Computations in association schemes, Ph.D. thesis, Eindhoven University of Technology, 1998.
L. J. Rylands, A formula for signs of structure constants, Unpublished, 2001.
Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Robert Steinberg, Générateurs, relations et revêtements de groupes algébriques, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) Librairie Universitaire, Louvain; Gauthier-Villars, Paris, 1962, pp. 113–127 (French). MR 0153677
Robert Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56. MR 155937
Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
M. A. A. van Leeuwen, A. M. Cohen, and B. Lisser, LiE manual, CWI/CAN, Amsterdam, 1992, Manual for the software package LiE for Lie group theoretical computations. http://young.sp2mi.univ-poitiers.fr/~marc/LiE/.