Orbit spaces of reflection groups with 2, 3 and 4 basic polynomial invariants

Abstract

Functions which are covariant or invariant under the transformations of a compact linear group can be expressed advantageously in terms of functions defined in the orbit space of the group, i.e. as functions of a finite set of basic invariant polynomials. The equalities and inequalities defining the orbit spaces of all finite coregular real linear groups (most of which are crystallographic groups) with at most four independent basic invariants are determined. For each group G acting in the Euclidean space , the results are obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants of G; the semi-positivity conditions are known to determine all the equalities and inequalities defining the orbit space of G as a semi-algebraic variety in the space spanned by the variables . In a recent paper, the -matrices, for , have been determined in an alternative way, as solutions of a universal differential equation; the present paper yields a partial, but significant, check on the correctness and completeness of these solutions. Our results can be easily exploited, in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid-state theory where use is made of symmetry adapted functions.