P-matrices in orbit spaces and invariant theory

In many physical problems or applications one has to study functions that are invariant under the action of a symmetry group G and this is best done in the orbit space of G if one knows the equations and inequalities defining the orbit space and its strata. It is reviewed how the P̂-matrix is defined in terms of an integrity basis and how it can be used to determine the equations and inequalities defining the orbit space and its strata. It is shown that the P̂-matrix is a useful tool of constructive invariant theory, in fact, when the integrity basis is only partially known, calculating the P̂-matrix elements, one is able to determine the integrity basis completely.