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

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308021-4

MathSciNet review:
1308021

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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****[2]**Arjeh M. Cohen and David B. Wales,*Finite subgroups of 𝐺₂(𝐶)*, Comm. Algebra**11**(1983), no. 4, 441–459. MR**689418**, https://doi.org/10.1080/00927878308822857**[3]**-,*Finite subgroups of**and*, 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**, https://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****[7]**Larry Dornhoff,*Group representation theory. Part A: Ordinary representation theory*, Marcel Dekker, Inc., New York, 1971. Pure and Applied Mathematics, 7. MR**0347959****[8]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[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****[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**, https://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**, https://doi.org/10.2307/2372999**[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-New York, 1982, pp. 157–165. MR**675113****[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****[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,*and the rank two Lie groups: their construction, geometry, and invariants in light of Kostant's Conjecture*, Ph.D. thesis, MIT, May, 1994.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308021-4

Keywords:
,
,
Kostant's conjecture

Article copyright:
© Copyright 1995
American Mathematical Society