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

TL;DR

This paper characterizes the orbit spaces of finite coregular real linear groups with up to four invariants, providing algebraic relations and a method to verify their structure through metric matrices, aiding various symmetry-related fields.

Contribution

It determines the semialgebraic relations defining orbit spaces for groups with up to four invariants and verifies solutions obtained via differential equations, advancing understanding of group invariants.

Findings

Derived semialgebraic relations for orbit spaces with up to 4 invariants.

Validated solutions for metric matrices using differential equations.

Facilitated applications in symmetry breaking and phase transition analysis.

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 $\real^n$, the results are obtained through the computation of a metric matrix $\widehat P(p)$, 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 $\widehat P(p)\ge 0$ are known to determine all the equalities and inequalities defining the orbit space $\real^n/G$ of $G$ as a semi-algebraic variety in the space $\real^q$ spanned by the variables $p_1,\dots ,p_q$. In a recent paper, the $\widehat P$-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.