Constructing representations of higher degrees of finite simple groups and covers

Author:
Vahid Dabbaghian-Abdoly

Journal:
Math. Comp. **76** (2007), 1661-1668

MSC (2000):
Primary 20C40; Secondary 20C15

DOI:
https://doi.org/10.1090/S0025-5718-07-01969-2

Published electronically:
January 25, 2007

MathSciNet review:
2299793

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a finite group and $\chi$ an irreducible character of $G$. A simple method for constructing a representation affording $\chi$ can be used whenever $G$ has a subgroup $H$ such that $\chi _H$ has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree between 32 and 100, then such a subgroup exists.

- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - ATLAS of Finite Group Representations, School of Mathematics and Statistics, The University of Birmingham, Version 2, (http://web.mat.bham.ac.uk/atlas/v2.0/).
- Vahid Dabbaghian-Abdoly,
*Constructing representations of finite simple groups and covers*, Canad. J. Math.**58**(2006), no. 1, 23–38. MR**2195590**, DOI https://doi.org/10.4153/CJM-2006-002-3 - V. Dabbaghian-Abdoly,
*An algorithm to construct representations of finite groups*, Ph.D. thesis, School of Mathematics, Carleton University, 2003. - V. Dabbaghian-Abdoly,
*RPSEN - A Package for Constructing Representations of Finite Groups*, GAP Package, 2004, (http://www.gap-system.org/Packages/repsn.html). - John D. Dixon,
*Constructing representations of finite groups*, Groups and computation (New Brunswick, NJ, 1991) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 11, Amer. Math. Soc., Providence, RI, 1993, pp. 105–112. MR**1235797** - The GAP Group, GAP—Groups, Algorithms, and Programming. Version 4.6 (2005), (http://www.gap-system.org).
- R. Gow,
*Schur indices of some groups of Lie type*, J. Algebra**42**(1976), no. 1, 102–120. MR**466330**, DOI https://doi.org/10.1016/0021-8693%2876%2990029-6 - Erhan Güzel,
*Primitive idempotents of the group algebra ${\bf C}{\rm SL}(3,q)$*, Math. Scand.**70**(1992), no. 2, 177–185. MR**1189972**, DOI https://doi.org/10.7146/math.scand.a-12394 - G. J. Janusz,
*Primitive idempotents in group algebras*, Proc. Amer. Math. Soc.**17**(1966), 520–523. MR**194523**, DOI https://doi.org/10.1090/S0002-9939-1966-0194523-0 - Daniel Gorenstein, Richard Lyons, and Ronald Solomon,
*The classification of the finite simple groups. Number 3. Part I. Chapter A*, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR**1490581** - I. Martin Isaacs,
*Character theory of finite groups*, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR**1280461** - Gregory Karpilovsky,
*The Schur multiplier*, London Mathematical Society Monographs. New Series, vol. 2, The Clarendon Press, Oxford University Press, New York, 1987. MR**1200015** - Zyozyu Ohmori,
*On a Zelevinsky theorem and the Schur indices of the finite unitary groups*, J. Math. Sci. Univ. Tokyo**4**(1997), no. 2, 417–433. MR**1466354** - Neşe Yelkenkaya,
*Primitive idempotents of the group algebra ${\rm CSL}(2,q)$*, İstanbul Üniv. Fen Fak. Mat. Derg.**55/56**(1996/97), 99–109 (2000). MR**1767540** - Andrey V. Zelevinsky,
*Representations of finite classical groups*, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR**643482**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
20C40,
20C15

Retrieve articles in all journals with MSC (2000): 20C40, 20C15

Additional Information

**Vahid Dabbaghian-Abdoly**

Affiliation:
The Centre for Experimental and Constructive Mathematics (CECM), Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada

Email:
vdabbagh@cecm.sfu.ca

Keywords:
Simple group,
central cover,
irreducible representation

Received by editor(s):
November 27, 2005

Received by editor(s) in revised form:
July 6, 2006

Published electronically:
January 25, 2007

Additional Notes:
This work was supported by the MITACS NCE and NSERC of Canada

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.