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Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

Published online by Cambridge University Press:  01 February 2010

Frank Lübeck
Frank Lübeck
Affiliation:
RWTH Aachen, Lehrstuhl D für Mathematik, Templergraben 64, 52062 Aachen, Germany, [email protected], http://www.math.rwth-aachen.de/~Frank.Luebeck/

Abstract

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The author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 4 , 2001 , pp. 135 - 169
Copyright
Copyright © London Mathematical Society 2001

References

1.Benson, D. J., Representations and cohomology I, Cambr. Stud. Adv. Math. 30 (Cambridge University Press, 1991).Google Scholar
2.Burgoyne, N., ‘Modular representations of some finite groups’, Representation theory of finite groups and related topics, Proc. Symp. Pure Math. XXI (Amer. Math. Soc, Providence, RI, 1971) 13–17.Google Scholar
3.Carter, R.W., Simple groups of Lie type (Wiley-Interscience, London, 1972).Google Scholar
4.Curtis, C.W. and Reiner, I., Methods of representation theory, vols I and II (Wiley, New York, 1981/1987).Google Scholar
5.Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G., ‘CHEVIE–A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras’, Appl. Algebra Engrg. Comm. Comput. 7 (1996) 175–210.CrossRefGoogle Scholar
6.Gilkey, P.B. and Seitz, G.M., ‘Some representations of exceptional Lie algebras’, Geom. Dedicata 25 (1988) 407–416.Google Scholar
7.Hiss, G. and Malle, G., ‘Low-dimensional representations of quasi-simple groups’, LMS J. Comput. Math. 4 (2001) 22–63.CrossRefGoogle Scholar
8.Humphreys, J.E., Introduction to Lie algebras and representation theory (Springer, New York/Heidelberg/Berlin, 1972).CrossRefGoogle Scholar
9.Humphereys, J.E., ‘Modular representations in finite groups of Lie type’, Finite simple groups II (ed. Collins, M. J., Academic Press, London, 1980) 259–290.Google Scholar
10.Jansen, C., Lux, K., Parker, R. and Wilson, R., An atlas of Brauer characters (The Clarendon Press/Oxford University Press, New York, 1995).Google Scholar
11.Jantzen, J.C., ‘Darstellungen halbeinfacher Gruppen und kontravariante Formen’, J. Reine Angew. Math. 290 (1977) 117–141.Google Scholar
12.Jantzen, J.C., ‘Representations of algebraic groups’, Pure Appl. Math. 131 (Academic Press, London, 1987).Google Scholar
13.Kleidman, P. and Liebeck, M.The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser. 129 (Cambridge University Press, 1990).CrossRefGoogle Scholar
14.Lübeck, F., ‘On the computation of elementary divisors of integer matrices’, J. Sym bolic Comput., to appear.Google Scholar
15.Moody, R.V. and Patera, J., ‘Fast recursion formula for weight multiplicities’, Bull. Amer. Math. Soc. (N.S.) 1 (1982) 237–242.Google Scholar
16.Premet, A.A., ‘Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic’, Math. USSR Sb. 61 (1988) 167–183 (translated from the Russian original in Math. Sb. (N.S.)).CrossRefGoogle Scholar
17.Schönert, Martin et al. , GAP-groups, algorithms and programming, 3rd edn (Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1993–1997).Google Scholar
18.Steinberg, R., ‘Lectures on Chevalley groups’ (Yale University, 1967).Google Scholar
19.Steinberg, R., ‘Endomorphisms of linear algebraic groups’, Mem. Amer. Math. Soc. 80 (1968).Google Scholar