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

## TL;DR

This paper investigates the regularity of invariant functions under finite reflection groups, establishing conditions under which such functions can be expressed as functions of invariants with specific differentiability classes.

## Contribution

It characterizes the space of invariant functions that can be written as functions of invariants with a given smoothness and identifies when the composition achieves higher differentiability.

## Key findings

- Determines when invariant functions of class C^{hr} can be expressed as functions of invariants of class C^r.
- Identifies the subspace of functions F of class C^r for which F∘P is of class C^{hr}.
- Provides a detailed analysis of differentiability properties in the context of finite reflection groups.

## Abstract

Let $W$ be a finite reflection group. A $W$-invariant function of class~$C^{\infty}$ may be expressed as a functions of class $C^{\infty}$ of the basic invariants. In finite class of differentiability, the situation is not this simple. Let~$h$ be the greatest Coxeter number of the irreducible components of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariant function of class $C^{hr}$, and $F$ is the function of invariants associated by $f=F\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, in general $f=F\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here we determine the space of $W$-invariant functions that may be written as functions of class $C^r$ of the polynomial invariants and the subspace of functions $F$ of class $C^r$ of the invariants such that the invariant function $f=F\circ P$ is of class $C^{hr}$.