Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Weyl groups and the regularity properties of certain compact Lie group actions
HTML articles powered by AMS MathViewer

by Eldar Straume PDF
Trans. Amer. Math. Soc. 306 (1988), 165-190 Request permission

Abstract:

The geometric weight system of a $G$-manifold $X$ (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given $G$-weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the $G$-weight pattern must in fact coincide with that of a simple representation. This in turn implies that $X$ is (orthogonally) modeled on the linear $G$-space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of $G$-manifolds, $G$ a classical group. In this family the weight patterns are of $2$-parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.
References
  • Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • Wu-chung Hsiang and Wu-yi Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705–786. MR 217213, DOI 10.2307/2373241
  • Wu-yi Hsiang, On the splitting principle and the geometric weight system of topological transformation groups. I, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 334–402. MR 0380845
  • Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384
  • Wu-yi Hsiang and Eldar Straume, Actions of compact connected Lie groups with few orbit types, J. Reine Angew. Math. 334 (1982), 1–26. MR 667447, DOI 10.1515/crll.1982.334.1
  • Wu-Yi Hsiang and Eldar Straume, Actions of compact connected Lie groups on acyclic manifolds with low-dimensional orbit spaces, J. Reine Angew. Math. 369 (1986), 21–39. MR 850627
  • Wu-Yi Hsiang and Eldar Straume, On the orbit structures of $\textrm {SU}(n)$-actions on manifolds of the type of Euclidean, spherical or projective spaces, Math. Ann. 278 (1987), no. 1-4, 71–97. MR 909218, DOI 10.1007/BF01458061
  • E. Straume, $p$-weights and their application to regular actions of classical groups, Math. Rep., Univ. of Oslo, 1975.
  • Eldar J. Straume, Dihedral transformation groups of homology spheres, J. Pure Appl. Algebra 21 (1981), no. 1, 51–74. MR 609271, DOI 10.1016/0022-4049(81)90075-X
  • Eldar Straume, The topological version of groups generated by reflections, Math. Z. 176 (1981), no. 3, 429–446. MR 610222, DOI 10.1007/BF01214618
  • P. Yang, Adjoint type representations of classical groups on homology spheres, Thesis, Princeton Univ., Princeton, N.J., 1974.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 57S15, 20G05
  • Retrieve articles in all journals with MSC: 57S15, 20G05
Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 306 (1988), 165-190
  • MSC: Primary 57S15; Secondary 20G05
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0927687-8
  • MathSciNet review: 927687