of functions which are equivariant or invariant under the transformations of a compact linear group (hereafter abbreviated in CLG) $G$ is often a basic problem to solve in many physical contexts. $G$-invariant functions play an important role, for instance, in the determination of patterns of spontaneous symmetry breaking and structural phase transitions (Landau theory [@470; @620; @051; @557]), in equivariant bifurcation theory (see, for instance, [@Chossat] and references therein), in crystal field theory and in most areas of solid state theory.

It will not be essentially restrictive, in the following, to assume that $G$ is a matrix subgroup of the real group O(n). In fact, complex linear groups can always be transformed into real linear groups through a process of realification and compact real linear groups are equivalent to orthogonal linear groups.

An approach to the study of invariant functions, that fully exploits invariance and possible regularity properties, takes advantage of the fact that a $G$-invariant function $f$, defined on $\real^n$, is a constant along each orbit of $G$ and can, therefore, be considered as a function $\widehat f$ in the orbit space $\real^n/G$ of the action of $G$ in $\real^n$. Geometric invariant theory [@610; @611] suggests how to get, 