Orbit Spaces: A tool to analyze phase transitions

In the framework of the Landau theory of phase transitions one is interested to describe all the possible low symmetry phases allowed for a given physical system and to determine the conditions under which the system undergoes a phase transition. These problems are best described and analyzed in the orbit space of the high symmetry group of the system but one has to know exactly the structure of this orbit space. Using some results of invariant theory it is possible to determine in a clear and simple way all the equations and inequalities defining the orbit spaces of every compact linear group. In this article are reviewed the main properties of the orbit spaces of compact linear groups concerning Landau theory of phase transitions and the mathematical techniques used to determine the equations and inequalities defining the orbit spaces and their stratifications. Some examples concerning phase transitions in superconducting systems are given.