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Reflection quotients in Riemannian geometry. A geometric converse to Chevalley's theorem
Author(s):
R.
Milson
Journal:
Proc. Amer. Math. Soc.
132
(2004),
2825-2831.
MSC (2000):
Primary 20H15, 14L24, 53B21
Posted:
June 2, 2004
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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:
- 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
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Additional Information:
R.
Milson
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email:
milson@mscs.dal.ca
DOI:
10.1090/S0002-9939-04-07583-5
PII:
S 0002-9939(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
Posted:
June 2, 2004
Additional Notes:
The author was supported by NSERC grant 228057
Communicated by:
Wolfgang Ziller
Copyright of article:
Copyright
2004,
American Mathematical Society
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