Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata

TL;DR

This paper introduces a straightforward method to obtain rational parametrizations of strata in the orbit space of a compact linear group, facilitating the analysis of invariant functions in various physical and mathematical contexts.

Contribution

It provides a simple recipe for rationally parametrizing strata in the orbit space, improving upon heuristic methods and aiding studies involving symmetry-invariant functions.

Findings

The recipe simplifies the parametrization process for strata.

Application to liquid crystals demonstrates practical advantages.

Method enhances analysis in symmetry-related physical theories.

Abstract

Functions which are equivariant or invariant under the transformations of a compact linear group $G$ acting in an euclidean space $\real^n$, can profitably be 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 orbit-type. 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 equivariant 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 theory), in equivariant bifurcation theory, in crystal field theory and in most areas where use is made of symmetry adapted functions. A physically significant example of utilization of the recipe is given, related to spontaneous polarization in chiral biaxial liquid crystals, where the advantages with respect to previous heuristic approaches are shown.