Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T05:54:01.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-Dimensional Representations of Quasi-Simple Groups

Published online by Cambridge University Press:  01 February 2010

Gerhard Hiss  and
Gunter Malle
Gerhard Hiss
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany, Gerhard.Hiss@Math.RWTH-Aachen.DE
Gunter Malle
Affiliation:
FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany, malle@mathematik.uni-kassel.de

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 authors determine all the absolutely irreducible representations of degree up to 250 of quasi-simple finite groups, excluding groups that are of Lie type in their defining characteristic. Additional information is also given on the Frobenius-Schur indicators and the Brauer character fields of the representations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 4 , 2001 , pp. 22 - 63
Copyright
Copyright © London Mathematical Society 2001

References

1.Benson, D., ‘Some remarks on the decomposition numbers for the symmetric groups’, Proc. Symp. Pure Math. 47 (1987) 381–394.CrossRefGoogle Scholar
2.Benson, D., ‘Spin modules for symmetric groups’, J. London Math. Soc. 38 (1988) 250–262.CrossRefGoogle Scholar
3.Burkhardt, R., ‘Die Zerlegungsmatrizen der Gruppen PSL(2, pƒ)’, J. Algebra 40 (1976) 75–96.CrossRefGoogle Scholar
4.Char, B. W. et al. , Maple V, language reference manual (Springer, Berlin, 1991).CrossRefGoogle Scholar
5.Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
6.Enomoto, H., ‘The characters of the finite symplectic group Sp(4, q), q = 2ƒ’, Osaka J. Math. 9 (1972) 75–94.Google Scholar
7.Fong, P. and Srinivasan, B., ‘Brauer trees in classical groups’, J. Algebra 131 (1990) 179–225.CrossRefGoogle Scholar
8.THE GAP GROUP, GAP—groups, algorithms, and programming (The GAP Group, Aachen/St Andrews, 1999); http://www-gap.dcs.st-and.ac.uk/gap.Google Scholar
9.Geck, M., ‘Generalized Gelfand-Graev characters for Steinberg's triality groups and their applications’, Comm. Algebra 19 (1991) 3249–3269.Google Scholar
10.Gow, R., ‘Schur indices of some groups of Lie type’, J. Algebra 42 (1976) 102–120.CrossRefGoogle Scholar
11.Gow, R., ‘On the Schur indices of characters of finite classical groups’, J. London Math. Soc. 24 (1981) 135–147.CrossRefGoogle Scholar
12.Gow, R., ‘Contraction of exterior powers in characteristic 2 and the spin module’, Geom. Dedicata 64 (1997) 283–295.CrossRefGoogle Scholar
13.Gow, R. and Willems, W., ‘Methods to decide if simple self-dual modules over fields of characteristic 2 are of quadratic type’, J. Algebra 175 (1995) 1067–1081.CrossRefGoogle Scholar
14.Gow, R. and Willems, W., ‘On the quadratic type of some simple self-dual modules over fields of characteristic two’, J. Algebra 195 (1997) 634–649.CrossRefGoogle Scholar
15.Guralnick, R. and Tiep, P. H., ‘Low-dimensional representations of special linear groups in cross characteristics’, Proc. London Math. Soc. 78 (1999) 116–138.CrossRefGoogle Scholar
16.Guralnick, R., Magaard, K., Saxl, J. and TIEP, P. H., ‘Cross characteristic representations of odd characteristic symplectic groups and unitary groups’, Preprint, http://math.usc.edurguralnic/Preprints/qmstu.dviGoogle Scholar
17.Henke, A., Hiss, G. and Müller, J., ‘The 7-modular decomposition matrices of the sporadic O'Nan group’, J. London Math. Soc. 60 (1999) 58–70.CrossRefGoogle Scholar
18.Hensing, S., ‘5-modulare Zerlegungszahlen der sporadischen einfachen Gruppe Colund ihrer Überlagerungsgruppe 2.Col’, Diplomarbeit, Universität Heidelberg, 1993.Google Scholar
19.Hiss, G., ‘The 3-modular characters of the Rudvalis sporadic simple group and its covering group’, Math. Comp. 62 (1997) 851–863.CrossRefGoogle Scholar
20.Hiss, G., ‘Decomposition matrices of the Chevalley group F4(2) and its covering group’, Comm. Algebra 25 (1997) 2539–2555.CrossRefGoogle Scholar
21.Hiss, G. and Lux, K., Brauer trees of sporadic groups (Clarendon Press, Oxford, 1989).Google Scholar
22.Hiss, G. and Lux, K., ‘The 5-modular characters of the sporadic simple Fischer groups Fi22 and Fi23’, Comm. Algebra 22 (1994) 3563–3590.CrossRefGoogle Scholar
23.Hiss, G. and Malle, G., ‘Low-dimensional representations of special unitary groups’, J. Algebra 236 (2001) 745–767.CrossRefGoogle Scholar
24.Hiss, G. and Müller, J., ‘The 5-modular characters of the sporadic simple Rudvalis group and its covering group’, Comm. Algebra 23 (1995) 4633–4667.CrossRefGoogle Scholar
25.Hiss, G. and White, D. L., ‘The 5-modular characters of the covering group of the sporadic simple Fischer group Fi22 and its automorphism group’, Comm. Algebra 22 (1994) 3591–3611.CrossRefGoogle Scholar
26.Isaacs, M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
27.James, G. D., ‘Representations of the symmetric groups over the field of order 2’, J. Algebra 38 (1976) 280–308.CrossRefGoogle Scholar
28.James, G. D., 'The representation theory of the symmetric groups (Springer, Berlin, 1978).CrossRefGoogle Scholar
29.James, G. D., ‘The decomposition matrices of GLn(q) for n ≤ 10’, Proc. London Math. Soc. (3) 60 (1990) 225–265.CrossRefGoogle Scholar
30.Jansen, C., Ein Atlas 3-modularer Charaktertafeln (M + M Wissenschaftsverlag, Krefeld, 1995).Google Scholar
31.Jansen, C. and Müller, J., ‘The 3-modular decomposition numbers of the sporadic simple Suzuki group’, Comm. Algebra 25 (1997) 2437–2458.CrossRefGoogle Scholar
32.Jansen, C. and Wilson, R. A., ‘The 2-modular and 3-modular decomposition numbers for the sporadic simple O'Nan group and its triple cover’, J. London Math. Soc. 57 (1998) 71–90.CrossRefGoogle Scholar
33.Jansen, C., Hiss, G., Lux, K. and Parker, R., ‘Computational modular character theory’, Preprint, http://www.math.rwth-aachen.de/~MOC/CoMoChaT/Google Scholar
34.Jansen, C., Lux, K., Parker, R. and Wilson, R., An atlas of Brauer characters (Clarendon Press, Oxford, 1995).Google Scholar
35.Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
36.Kondratiev, A. S., ‘Finite linear groups of small degree', The atlas of finite groups: ten years on (ed. Curtis, R. T. and Wilson, R. A., Cambridge University Press, Cambridge, 1998) 139–148.CrossRefGoogle Scholar
37.Lübeck, F., ‘Small degree representations of finite Chevalley groups in defining characteristic’, Preprint, http://www.math.rwth-aachen.de/~Frank.Luebeck/preprints/smdegdefchar.dviGoogle Scholar
38.Lux, K. and Pahlings, H., ‘Computational aspects of representation theory of finite groups', Representation theory of finite groups and finite-dimensional algebras (ed. Michler, G. O. and Ringel, C. M., Birkhaüser, Basel, 1991) 37–64.CrossRefGoogle Scholar
39.Lux, K. and Pahlings, H., ‘Computational aspects of representation theory of finite groups, II', Algorithmic algebra and number theory (ed. Matzat, B. Heinrich et al. , Springer, Berlin, 1999) 381–397.CrossRefGoogle Scholar
40.Lux, K. and Wiegelmann, M., ‘Condensing tensor product modules', The atlas of finite groups: ten years on (ed. Curtis, R. T. and Wilson, R. A., Cambridge University Press, Cambridge, 1998) 174–190.CrossRefGoogle Scholar
41.Meyer, W., Neutsch, W. and Parker, R., ‘The minimal 5-representation of Lyons' sporadic group’, Math. Ann. 272 (1985) 29–39.CrossRefGoogle Scholar
42.THE MOC GROUP, MOC—modular characters (The MOC Group, Aachen, 1999); http://www.math.rwth-aachen.de/LDFM/homes/MOC/.Google Scholar
43.Müller, J., ‘The 5-modular decomposition matrix of the sporadic simple Conway group Co3', Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ISSAC ‘98, Rostock, Germany, August 13–15,1998 (ed. Gloor, O., Academic Press, New York, 1998) 179–185.Google Scholar
44.Nozawa, S., ‘On the characters of the finite general unitary group U(4, q2)’, J. Fac. Sci. Univ. Tokyo 19 (1972) 257–293Google Scholar
45.Okuyama, T. and Waki, K., ‘Decomposition numbers of Sp(4, q)’, J. Algebra 199(1998) 544–555.CrossRefGoogle Scholar
46.Przygocki, A., ‘Schur indices of symplectic groups’, Comm. Algebra 10 (1982) 279–310.CrossRefGoogle Scholar
47.Ryba, A. J. E., ‘Calculation of the 7-modular characters of the Held group’, J. Algebra 117 (1988) 240–255.CrossRefGoogle Scholar
48.Ryba, A. J. E., ‘Computer condensation of modular representations’, J. Symbolic Comput. 9 (1990) 591–600.CrossRefGoogle Scholar
49.Seitz, G. M. and Zalesskii, A. E., ‘On the minimal degrees of projective representations of the finite Chevalley groups, II’, J. Algebra 158 (1993) 233–243.CrossRefGoogle Scholar
50.Srinivasan, B., ‘The characters of the finite symplectic group Sp(4, q)’, Trans. Amer. Math. Soc. 131 (1968) 488–525.Google Scholar
51.Suleiman, I. A. I. and Wilson, R. A., ‘The 2-modular characters of Conway's group Co2’, Math. Proc. Cambridge Philos. Soc. 116 (1994) 275–283.CrossRefGoogle Scholar
52.Suleiman, I. A. I. and Wilson, R. A., ‘The 2-modular characters of Conway's third group Co3’, J. Symbolic Comput. 24 (1996) 493–506.CrossRefGoogle Scholar
53.Thompson, J., ‘Bilinear forms in characteristic p and the Frobenius-Schur indicator', Group Theory, Beijing 1984 (ed. Tuan, Hsio-Fu, Springer, Berlin, 1986) 221–230.Google Scholar
54.Tiep, P. H., ‘Weil representations as globally irreducible representations', Math. Nachr. 184 (1997) 313–327.Google Scholar
55.Wales, D. B., ‘Some projective representations of Sn’, J. Algebra 61 (1979) 37–57.CrossRefGoogle Scholar
56.White, D. L., ‘The 2-decomposition numbers of Sp(4, q), q odd’, J. Algebra 131 (1990) 703–725.CrossRefGoogle Scholar
57.White, D. L., ‘Decomposition numbers of Sp(4, q) for primes dividing q ± 1’, J. Algebra 132 (1990) 488–500.CrossRefGoogle Scholar
58.White, D. L., ‘Decomposition numbers of Sp4(2a) in odd characteristics’, J. Algebra 177 (1995) 264–276.CrossRefGoogle Scholar
59.Willems, W., ‘Metrische Moduln über Gruppenringen’, Dissertation, Universität Mainz, Germany, 1976.Google Scholar
60.Willems, W., ‘Duality and forms in representation theory’, Representation theory of finite groups and finite-dimensional algebras, (ed. Michler, G. O. and Ringel, C. M., Birkhäuser, Basel, 1991) 509–520.CrossRefGoogle Scholar
61.Wings, E., ‘Die Zerlegungszahlen der Hauptblöcke der unitären Gruppen GU4q2) in nicht-definierender Charakteristik’, Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1990.Google Scholar
Supplementary material: PDF

JCM 4 Hiss and Malle Appendix A

Hiss and Malle Appendix A

Download JCM 4 Hiss and Malle Appendix A(PDF)
PDF 114.5 KB