Orbit Spaces of Compact Linear Groups

TL;DR

This paper reviews the P-matrix approach for determining orbit spaces of compact linear groups, providing classifications for groups with up to four invariants and discussing implications for symmetry breaking and phase transitions.

Contribution

It offers a comprehensive review of the P-matrix method and provides classifications of orbit spaces for specific classes of compact linear groups, expanding understanding without requiring group structure details.

Findings

Classified orbit spaces for groups with up to 4 invariants.

Discussed properties of orbit spaces of coregular groups.

Linked orbit space stratification to symmetry breaking schemes.

Abstract

The P-matrix approach for the determination of the orbit spaces of compact linear groups enabled to determine all orbit spaces of compact coregular linear groups with up to 4 basic polynomial invariants and, more recently, all orbit spaces of compact non-coregular linear groups with up to 3 basic invariants. This approach does not involve the knowledge of the group structure of the single groups but it is very general, so after the determination of the orbit spaces one has to determine the corresponding groups. In this article it is reviewed the main ideas underlying the P-matrix approach and it is reported the list of linear irreducible finite groups and of linear compact simple Lie groups, with up to 4 basic invariants, together with their orbit spaces. Some general properties of orbit spaces of coregular groups are also discussed. This article will deal only with the mathematical aspect, however one must keep in mind that the stratification of the orbit spaces represents the possible schemes of symmetry breaking and that the phase transitions appear when the minimum of an invariant potential function shifts from one stratum to another, so the exact knowledge of the orbit spaces and their stratifications might be useful to single out some yet hidden properties of phase transitions.