Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T14:57:34.811Z Has data issue: false hasContentIssue false
  • English
  • Français

RECOGNITION OF SMALL DIMENSIONAL REPRESENTATIONS OF GENERAL LINEAR GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 October 2008

KAY MAGAARD ,
E. A. O’BRIEN  and
ÁKOS SERESS
KAY MAGAARD
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK (email: k.magaard@bham.ac.uk)
E. A. O’BRIEN*
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand (email: obrien@math.auckland.ac.nz)
ÁKOS SERESS
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA (email: akos@math.ohio-state.edu)
*
✉For correspondence; e-mail: obrien@math.auckland.ac.nz
Rights & Permissions [Opens in a new window]

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.

Let G be isomorphic to a group H satisfying SL(d,q)≤H≤GL(d,q) and let W be an irreducible FqG-module of dimension at most d2. We present a Las Vegas polynomial-time algorithm which takes as input W and constructs a d-dimensional projective representation of G.

Keywords

matrix recognition projectsymmetric squarealternating square adjoint representation

MSC classification

Secondary: 20C20: Modular representations and characters 20C40: Computational methods
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 229 - 250
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

This work was supported in part by the NSA, the Marsden Fund of New Zealand, and the NSF.

References

[1]Babai, László, ‘Local expansion of vertex-transitive graphs and random generation in finite groups’, in: Theory of Computing, Los Angeles, 1991 (Association for Computing Machinery, New York, 1991), pp. 164–174.Google Scholar
[2]Beals, Robert, Leedham-Green, Charles R., Niemeyer, Alice C., Praeger, Cheryl E. and Seress, Ákos, ‘A black-box group algorithm for recognizing finite symmetric and alternating groups I’, Trans. Amer. Math. Soc. 355 (2003), 2097–2113.CrossRefGoogle Scholar
[3]Beals, Robert, Leedham-Green, Charles R., Niemeyer, Alice C., Praeger, Cheryl E. and Seress, Ákos, ‘Constructive recognition of finite symmetric and alternating groups acting as matrix groups on their natural permutation modules’, J. Algebra 292 (2005), 4–46.CrossRefGoogle Scholar
[4]Bosma, Wieb, Cannon, John and Playoust, Catherine, ‘The Magma algebra system I: the user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
[5]Brooksbank, Peter A., ‘Constructive recognition of classical groups in their natural representation’, J. Symbolic Comput. 35 (2003), 195–239.CrossRefGoogle Scholar
[6]Celler, Frank, Leedham-Green, Charles R., Murray, Scott H., Niemeyer, Alice C. and O’Brien, E. A., ‘Generating random elements of a finite group’, Comm. Algebra 23 (1995), 4931–4948.CrossRefGoogle Scholar
[7] The Group. — Groups, Algorithms, and Programming. Version 4.4.9; 2007, (http://www.gap-system.org).Google Scholar
[8]Glasby, S. P., Leedham-Green, C. R. and O’Brien, E. A., ‘Writing projective representations over subfields’, J. Algebra 295 (2006), 51–61.CrossRefGoogle Scholar
[9]Kantor, William M. and Seress, Ákos, ‘Black box classical groups’, Mem. Amer. Math. Soc. 149 (2001).Google Scholar
[10]Keller-Gehrig, W., ‘Fast algorithms for the characteristic polynomial’, Theoret. Comput. Sci. 36 (1985), 309–317.CrossRefGoogle Scholar
[11]Leedham-Green, C. R. and O’Brien, E. A., Constructive recognition of classical groups in odd characteristic. Preprint 2007.Google Scholar
[12]Liebeck, Martin W., ‘On the orders of maximal subgroups of the finite classical groups’, Proc. London Math. Soc. (3) 50 (1985), 426–446.CrossRefGoogle Scholar
[13]Nathanson, M. B., Elementary Methods in Number Theory (Springer, New York, 2000).Google Scholar
[14]Neumann, P. M. and Praeger, C. E., ‘A recognition algorithm for special linear groups’, Proc. London Math. Soc. 65 (1992), 555–603.CrossRefGoogle Scholar
[15]O’Brien, E. A., ‘Towards effective algorithms for linear groups’, in: Finite Geometries, Groups and Computation, (Pingree Park, Co, 2004) (De Gruyter, Berlin, 2006), pp. 163–190.CrossRefGoogle Scholar
[16]Pak, Igor, ‘The product replacement algorithm is polynomial’, in: 41st Ann. Symp. on Foundations of Computer Science, (Redondo Beach, CA, 2000) (IEEE Computer Society Press, Los Alamitos, CA, 2000), pp. 476–485.Google Scholar
[17]Ryba, Alexander J. E., ‘Identification of matrix generators of a Chevalley group’, J. Algebra 309 (2007), 484–496.CrossRefGoogle Scholar
[18]Seress, Ákos, Permutation Group Algorithms, Cambridge Tracts in Mathematics, 152 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[19]Gathen, Joachim von zur and Gerhard, Jürgen, Modern Computer Algebra (Cambridge University Press, Cambridge, 2002).Google Scholar