Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Reflection quotients in Riemannian geometry. A geometric converse to Chevalley's theorem


Author: R. Milson
Journal: Proc. Amer. Math. Soc. 132 (2004), 2825-2831
MSC (2000): Primary 20H15, 14L24, 53B21
DOI: https://doi.org/10.1090/S0002-9939-04-07583-5
Published electronically: June 2, 2004
MathSciNet review: 2063099
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Chevalley's theorem and its converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that, in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely, that a freely-generated polynomial subring is closed with respect to the gradient product.


References [Enhancements On Off] (What's this?)

  • 1. Arnol'd, V.I., Indexes of singular points of 1-forms on manifolds with boundary, convolutions of invariants of groups generated by reflections, and singular projections of smooth surfaces. Uspekhi Mat. Nauk 34 (1979), pp. 3-38. MR 81e:58041
  • 2. Chevalley, C., Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), pp. 778-782. MR 17:345d
  • 3. Flato, L., Invariants of finite reflection groups. Enseign. Math. 24 (1978), pp. 237-292. MR 80j:14010
  • 4. Hermann, R. , Geodesics of singular Riemannian metrics. Bull. Amer. Math. Soc. 79 (1973), pp. 780-782. MR 50:3271
  • 5. Milson, R., Imprimitively generated Lie-algebraic Hamiltonians and separation of variables. Canad. J. Math. 50 (1998), pp. 1298-1322. MR 2000a:81246
  • 6. Saito, K., T. Yano and J. Sekiguchi., On a certain generator system of the ring of invariants of a finite reflection group. Comm. Algebra 8 (1980), pp. 373-408. MR 81g:20095
  • 7. Shephard, G.C. and J.A. Todd., Finite unitary reflection groups. Canad. J. Math. 6 (1954), pp. 274-304. MR 15:600b
  • 8. Terng, C.L. , Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21 (1985), pp. 79-107. MR 87e:53095

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20H15, 14L24, 53B21

Retrieve articles in all journals with MSC (2000): 20H15, 14L24, 53B21


Additional Information

R. Milson
Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email: milson@mscs.dal.ca

DOI: https://doi.org/10.1090/S0002-9939-04-07583-5
Keywords: Reflection groups, invariants, degenerate metrics
Received by editor(s): December 3, 2001
Received by editor(s) in revised form: June 12, 2002
Published electronically: June 2, 2004
Additional Notes: The author was supported by NSERC grant 228057
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society