The Determination of the Orbit Spaces of Compact Coregular Linear Groups

TL;DR

This paper presents a mathematical method for determining the equations defining the orbit spaces of compact coregular linear groups, aiding the study of phase transitions through symmetry analysis.

Contribution

It introduces a matrix differential equation approach to find orbit space equations without requiring detailed group structure or integrity basis knowledge.

Findings

Method successfully applied to 2, 3, and 4-dimensional orbit spaces.

Provides induction rules for higher-dimensional cases.

Establishes a link between orbit space determination and invariant theory.

Abstract

Some aspects of phase transitions can be more conveniently studied in the orbit space of the action of the symmetry group. After a brief review of the fundamental ideas of this approach, I shall concentrate on the mathematical aspect and more exactly on the determination of the equations defining the orbit space and its strata. I shall deal only with compact coregular linear groups. The method exposed has been worked out together with prof. G. Sartori and it is based on the solution of a matrix differential equation. Such equation is easily solved if an integrity basis of the group is known. If the integrity basis is unknown one may determine anyway for which degrees of the basic invariants there are solutions to the equation, and in all these cases also find out the explicit form of the solutions. The solutions determine completely the stratification of the orbit spaces. Such calculations have been carried out for 2, 3 and 4-dimensonal orbit spaces. The method is of general validity but the complexity of the calculations rises tremendously with the dimension $q$ of the orbit space. Some induction rules have been found as well. They allow to determine easily most of the solutions for the $(q+1)$-dimensional case once the solutions for the $q$-dimensional case are known. The method exposed is interesting because it allows to determine the orbit spaces without using any specific knowledge of group structure and integrity basis and evidences a certain hidden and yet unknown link with group theory and invariant theory.