𝐿₂(𝑞) and the rank two Lie groups: their construction in light of Kostant’s conjecture

Sepanski, Mark R.

doi:10.1090/S0002-9947-1995-1308021-4

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

$L\sb 2(q)$ and the rank two Lie groups: their construction in light of Kostant's conjecture

Author: Mark R. Sepanski
Journal: Trans. Amer. Math. Soc. 347 (1995), 3983-4021
MSC: Primary 20D06; Secondary 17B20, 22E60
MathSciNet review: 1308021
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with certain aspects of a conjecture made by B. Kostant in 1983 relating the Coxeter number to the occurrence of the simple finite groups in simple complex Lie groups. A unified approach to Kostant's conjecture that yields very general results for the rank two case is presented.

[1] Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344 (86i:22023)
[2] Arjeh M. Cohen and David B. Wales, Finite subgroups of 𝐺₂(𝐶), Comm. Algebra 11 (1983), no. 4, 441–459. MR 689418 (85b:20010), http://dx.doi.org/10.1080/00927878308822857
[3] -, Finite subgroups of ${E_6}(\mathbb{C})$ and ${F_4}(\mathbb{C})$ , preprint, 1992.
[4] -, Finite simple subgroups of semisimple complex Lie groups--a survey, preprint, 1994.
[5] Arjeh M. Cohen, Robert L. Griess Jr., and Bert Lisser, The group 𝐿(2,61) embeds in the Lie group of type 𝐸₈, Comm. Algebra 21 (1993), no. 6, 1889–1907. MR 1215552 (94m:20041), http://dx.doi.org/10.1080/00927879308824659
[6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, 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 (88g:20025)
[7] Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory, Marcel Dekker Inc., New York, 1971. Pure and Applied Mathematics, 7. MR 0347959 (50 #458a)
[8] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842 (48 #2197)
[9] Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716 (92e:11001)
[10] Peter B. Kleidman and A. J. E. Ryba, Kostant’s conjecture holds for 𝐸₇:𝐿₂(37)<𝐸₇(𝐶), J. Algebra 161 (1993), no. 2, 535–540. MR 1247371 (94k:20025), http://dx.doi.org/10.1006/jabr.1993.1234
[11] Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 0114875 (22 #5693)
[12] -, A tale of two conjugacy classes, Colloquium Lecture of the Amer. Math. Soc., 1983.
[13] Arne Meurman, An embedding of 𝑃𝑆𝐿(2,13) in 𝐺₂(𝐶), Lie algebras and related topics (New Brunswick, N.J., 1981) Lecture Notes in Math., vol. 933, Springer, Berlin, 1982, pp. 157–165. MR 675113 (84b:20046)
[14] M. A. Naĭmark and A. I. Štern, Theory of group representations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 246, Springer-Verlag, New York, 1982. Translated from the Russian by Elizabeth Hewitt; Translation edited by Edwin Hewitt. MR 793377 (86k:22001)
[15] Mohammad Ali Najafi, Clifford algebra structure on the cohomology algebra of compact symmetric spaces, Master's thesis, MIT, February, 1979.
[16] Mark R. Sepanski, ${L_2}(q)$ and the rank two Lie groups: their construction, geometry, and invariants in light of Kostant's Conjecture, Ph.D. thesis, MIT, May, 1994.