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

Details 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 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

2000-08-02

arxiv.org/abs/hep-th/0008024v1

Quaderni CNR n. 54, Symmetry and Perturbation Theory - Workshop Torino 15-20/12/96, D. Bambusi and G. Gaeta Ed. (1998) p. 204-

Physics

High Energy Physics

High Energy Physics - Theory

37 pages, 2 figures, latex

Scientific paper

Invariant functions under the transformations of a compact linear group $G$ acting in $\real^n$ can be expressed in terms of functions defined in the orbit space of $G$. We develop a method to determine the isotropy classes of the orbit spaces of all the real linear groups whose integrity bases (IB) satisfy only one independent relation. The method is tested for IB's formed by 3 (independent) basic invariants. The result is obtained through a metric matrix $\hat P(p)$, defined only from the scalar products between the gradients of a minimal IB. We determine the matrices $\wP(p)$ from a universal differential equation, which satisfy new convenient additional conditions, which fit for the non-coregular case. Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and so on.

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