Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-27T14:14:30.955Z Has data issue: false hasContentIssue false
  • English
  • Français

Representations of Algebraic Groups

Published online by Cambridge University Press:  22 January 2016

Robert Steinberg
Robert Steinberg*
Affiliation:
Institute for Advanced Study, Princeton, University of California, Los Angeles
Rights & Permissions [Opens in a new window]

Extract

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.

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 33 - 56
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Brauer, R. and Nesbitt, C., On the modular representations of groups of finite order, U. of Toronto Studies (1937), 1–21.Google Scholar
[2] Brauer, R. and Nesbitt, C., On the modular characters of groups, Ann. of Math. 42 (1941), 556–590.CrossRefGoogle Scholar
[3] Chevalley, C., Sur certains groupes simples, Töhoku Math. J. 7 (1955), 14–66.Google Scholar
[4] Curtis, C. W., Representations of Lie algebras of classical type with applications to linear groups, J. Math, and Mech. 9 (1960), 307–326.Google Scholar
[5] Curtis, C. W., On projective representations of certain finite groups, Proc. Amer. Math. Soc. 11 (1960), 852–860.Google Scholar
[6] Curtis, C. W., On the dimensions of the irreducible modules of Lie algebras of classical type, Trans. Amer. Math. Soc. 96 (1960), 135–142.Google Scholar
[7] Dickson, L. E., The abstract form (two papers), Quart. J. Math. 38 (1907), 141–158.Google Scholar
[8] Hertzig, D., Forms of algebraic groups, Proc. Amer. Math. Soc. 12 (1961), 657–660.CrossRefGoogle Scholar
[9] Lang, S., Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.CrossRefGoogle Scholar
[10] Mark, C., Thesis, U. of Toronto.Google Scholar
[11] Ree, R., A family of simple groups associated with the simple Lie algebra of type (Fi), second paper: (G2), Amer. J. Math. 83 (1961), 401–420, 432–462.Google Scholar
[12] Rosenlicht, M., Some rationality questions on algebraic groups, Ann. di Mat. 43 (1957), 25–50.CrossRefGoogle Scholar
[13] Séminaire, C. Chevalley, Classification des Groupes de Lie Algébriques (two volumes), Paris (1956-8).Google Scholar
[14] Séminaire, J.P “Sophus Lie,” Paris (1954-5).Google Scholar
[15] Serre, J. P., Groupes algébtiques et corps de classes, Hermann, Paris (1959).Google Scholar
[16] Steinberg, R., A geometric approach, Trans. Amer. Math. Soc. 71 (1951), 274–282.Google Scholar
[17] Steinberg, R., Prime power representations of finite linear groups II, Can. J. Math. 9 (1957), 347–351.Google Scholar
[18] Steinberg, R., Finite reflection groups, Trans. Amer. Math. Soc. 91 (1959), 493–504.CrossRefGoogle Scholar
[19] Steinberg, R., Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875–891.CrossRefGoogle Scholar
[20] Steinberg, R., The simplicity of certain groups, Pacific J. Math. 10 (1960), 1039–1041.CrossRefGoogle Scholar
[21] Steinberg, R., Générateurs, relations et revêtements de groupes algébriques, Colloque sur la théorie des groupes algébriques, Bruxelles (1961).Google Scholar
[22] Suzuki, M., On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105–145.Google Scholar
[23] Tits, J., Sur les analogues algébriques des groupes semisimple complexes, Colloque d’algèbre supérieure, Bruxelles (1956), 261–289.Google Scholar
[24] Tits, J., Les “formes réelles” des groupes de type Ee, Séminaire Bourbaki 162, Paris (1958).Google Scholar
[25] Tits, J., Sur la trialité et certains groupes qui s’en déduisent, Paris Inst. Hautes Etudes Sci. Publ. Math. 2 (1959), 37–84.Google Scholar
[26] Weyl, H., Théorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transí or mat ionen III, Math. Zeit (1926), 377–395.CrossRefGoogle Scholar
[27] Wong, W. J., Thesis, Harvard U. Google Scholar