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Whitney regularity of the image of the Chevalley mapping

Part of: Infinite-dimensional manifolds Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 July 2016

Gérard P. Barbançon
Gérard P. Barbançon*
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA ([email protected])
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Extract

A compact set K ⊂ ℝn is Whitney 1-regular if the geodesic distance in K is equivalent to the Euclidean distance. Let P be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group. This paper gives the Whitney 1-regularity of the image by P of any closed ball centred at the origin. The proof uses the works of Givental', Kostov and Arnol'd on the symmetric group. It needs a generalization of a property of the Vandermonde determinants to the Jacobian of the Chevalley mappings.

Keywords

Whitney 1-regularityfinite reflection groupsMorse theoryfinite class of differentiability

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 58B10: Differentiability questions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 5 , October 2016 , pp. 895 - 904
Copyright
Copyright © Royal Society of Edinburgh 2016 

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