A canonical system of basic invariants of a finite reflection group

TL;DR

This paper introduces explicit formulas for canonical systems of basic invariants in finite reflection groups, linking their properties to invariant spaces and mean value properties, independent of group classification.

Contribution

It provides a classification-independent method to explicitly construct canonical systems of basic invariants for finite reflection groups.

Findings

Explicit formulas for canonical systems derived

Identification of invariant space with antiinvariant space

Construction does not rely on group classification

Abstract

A canonical system of basic invariants is a system of invariants satisfying a set of differential equations. The properties of a canonical system are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a canonical system of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of canonical systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups.