Rational parametrization of strata in orbit spaces of compact linear groups

TL;DR

This paper introduces a straightforward method for obtaining rational parametrizations of strata in orbit spaces of compact linear groups, facilitating the study of invariant and covariant functions in various physical and mathematical contexts.

Contribution

The authors present a simple recipe for rationally parametrizing strata in orbit spaces, aiding the analysis of symmetry-related functions across multiple scientific fields.

Findings

Provides an explicit method for rational parametrization of strata.

Enables easier analysis of invariant functions in physics and mathematics.

Includes an example demonstrating the application of the method.

Abstract

Functions which are covariant or invariant under the transformations of a compact linear group $G$ acting in a euclidean space $\real^n$, can be profitably studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same symmetry. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid state theory where use is made of symmetry adapted functions. An example of utilization of the recipe is also discussed at the end of the paper.