Finite Reflection Groups: Invariant functions and functions of the\n Invariants in finite class of differentiability

2019/06/15 by Gérard Barbançon, Barbançon, Gerard
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1906.06494

Abstract

Let W be a finite reflection group. A W-invariant function of\nclass~C^\∞ may be expressed as a functions of class C^\∞ of the\nbasic invariants. In finite class of differentiability, the situation is not\nthis simple. Let~h be the greatest Coxeter number of the irreducible\ncomponents of W and P be~the Chevalley mapping, if f is an invariant\nfunction of class C^hr, and F is the function of invariants associated by\nf=F\∘ P, then F is of class C^r. However if~F is of class C^r, in\ngeneral f=F\∘ P is of class C^r and not of class C^hr. Here we\ndetermine the space of W-invariant functions that may be written as functions\nof class C^r of the polynomial invariants and the subspace of functions F\nof class C^r of the invariants such that the invariant function f=F\∘ P\nis of class C^hr.\n

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