Reflection quotients in Riemannian geometry. A geometric converse to Chevalley's theorem
Author:
R. Milson
Journal:
Proc. Amer. Math. Soc. 132 (2004), 28252831
MSC (2000):
Primary 20H15, 14L24, 53B21
Published electronically:
June 2, 2004
MathSciNet review:
2063099
Fulltext PDF Free Access
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Abstract: Chevalley's theorem and its converse, the SheppardTodd 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 freelygenerated polynomial subring is closed with respect to the gradient product.
 1.
V.
I. Arnol′d, Indexes of singular points of 1forms on
manifolds with boundary, convolutions of invariants of groups generated by
reflections, and singular projections of smooth surfaces, Uspekhi Mat.
Nauk 34 (1979), no. 2(206), 3–38 (Russian). MR 535708
(81e:58041)
 2.
Claude
Chevalley, Invariants of finite groups generated by
reflections, Amer. J. Math. 77 (1955), 778–782.
MR
0072877 (17,345d)
 3.
Leopold
Flatto, Invariants of finite reflection groups, Enseign. Math.
(2) 24 (1978), no. 34, 237–292. MR 519546
(80j:14010)
 4.
Robert
Hermann, Geodesics of singular Riemannian
metrics, Bull. Amer. Math. Soc. 79 (1973), 780–782. MR 0350779
(50 #3271), http://dx.doi.org/10.1090/S000299041973133130
 5.
Robert
Milson, Imprimitively generated Liealgebraic Hamiltonians and
separation of variables, Canad. J. Math. 50 (1998),
no. 6, 1298–1322. MR 1657720
(2000a:81246), http://dx.doi.org/10.4153/CJM19980632
 6.
Kyoji
Saito, Tamaki
Yano, and Jirō
Sekiguchi, On a certain generator system of the ring of invariants
of a finite reflection group, Comm. Algebra 8 (1980),
no. 4, 373–408. MR 558611
(81g:20095), http://dx.doi.org/10.1080/00927878008822464
 7.
G.
C. Shephard and J.
A. Todd, Finite unitary reflection groups, Canadian J. Math.
6 (1954), 274–304. MR 0059914
(15,600b)
 8.
ChuuLian
Terng, Isoparametric submanifolds and their Coxeter groups, J.
Differential Geom. 21 (1985), no. 1, 79–107. MR 806704
(87e:53095)
 1.
 Arnol'd, V.I., Indexes of singular points of 1forms on manifolds with boundary, convolutions of invariants of groups generated by reflections, and singular projections of smooth surfaces. Uspekhi Mat. Nauk 34 (1979), pp. 338. MR 81e:58041
 2.
 Chevalley, C., Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), pp. 778782. MR 17:345d
 3.
 Flato, L., Invariants of finite reflection groups. Enseign. Math. 24 (1978), pp. 237292. MR 80j:14010
 4.
 Hermann, R. , Geodesics of singular Riemannian metrics. Bull. Amer. Math. Soc. 79 (1973), pp. 780782. MR 50:3271
 5.
 Milson, R., Imprimitively generated Liealgebraic Hamiltonians and separation of variables. Canad. J. Math. 50 (1998), pp. 12981322. 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. 373408. MR 81g:20095
 7.
 Shephard, G.C. and J.A. Todd., Finite unitary reflection groups. Canad. J. Math. 6 (1954), pp. 274304. MR 15:600b
 8.
 Terng, C.L. , Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21 (1985), pp. 79107. 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:
http://dx.doi.org/10.1090/S0002993904075835
PII:
S 00029939(04)075835
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
