Whitney Regularity of the Image of the Chevalley mapping

TL;DR

This paper proves Whitney 1-regularity of the image of closed balls under the Chevalley map associated with reflection groups, using advanced algebraic and geometric techniques.

Contribution

It establishes Whitney regularity for Chevalley map images, extending previous results and involving a novel generalization of Van der Monde determinant properties.

Findings

The image of closed balls under the Chevalley map is Whitney 1-regular.

The proof leverages works of Givental', Kostov, and Arnold on symmetric groups.

A new property of Van der Monde determinants is developed for Jacobian analysis.

Abstract

A closed set $F$ is Whitney 1-regular if for each compact $K\subset F$, 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 note gives the Whitney regularity of the image by $P$ of closed balls centered at the origin of ${\mathbb R}^n$. The proof uses the works of Givental', Kostov and Arnold on the symmetric group. It needs a generalization of a property of the Van der Monde determinants to the Jacobian of the Chevalley mappings.