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

Abstract

Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups with at most 4 basic invariants are determined. For each group $G$ acting in ${\Re }^n$, the results are obtained through the computation of a metric matrix $\hat{P}(p)$, which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants $p_1(x),\dots p_q(x),\x\in\real^n$ of $G$; the semi-positivity conditions $\hat{P}(p)\ge 0$ are known to determine all the equalities and inequalities defining the orbit space ${\Re }^n/G$ of $G$ as a semi-algebraic variety in the space ${\Re }^q$ spanned by the variables $p_1,\dots ,p_q$. In a recent paper, the $\hat{P}$-matrices, for $q\le 4$, 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 widely exploited,e.g. in the determination of patterns of spontaneous symmetry breaking, in the analysis of structural phase transitions (Landau's theory),in covariant bifurcation theory,in crystal field theory and in solid state theory where symmetry adapted functions are used.

42 pages, latex, accepted by Journal of Physics A, Mathematical and General

Cited by in corpus (6)

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• Michel theory of symmetry breaking and gauge theories
• Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space
• P-matrices in orbit spaces and invariant theory
• Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata
• Determination of the orbit spaces of non-coregular compact linear groups with one relation among the basic polynomial invariants in the $\hat{P}$-matrix approach