  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 P-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
P-matrix is a useful tool of constructive invariant theory, in fact, when the
integrity basis is only partially known, calculating the P-matrix elements, one
is able to determine the integrity basis completely.
