Dunkl Operators and Canonical Invariants of Reflection Groups

TL;DR

This paper introduces a new family of canonical invariants for finite reflection groups using Dunkl operators, demonstrating their relation to symmetric polynomials and hypergeometric functions for dihedral groups.

Contribution

It develops a continuous family of invariants via Dunkl operators and explicitly computes them for dihedral groups, extending the understanding of reflection group invariants.

Findings

Elementary canonical invariants deform elementary symmetric polynomials.

Canonical invariants for dihedral groups are expressed as hypergeometric functions.

The approach links Dunkl operators with classical invariant theory.

Abstract

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.