Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups
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Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups G. SARTORI and G. VALENTE Dipartimento di Fisica, Università di Padova and INFN, Sezione di Padova, I-35131 Padova, Italy Abstract. We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix (p), which is defined only in terms of the scalar products between the gradients ∂p1 (x), . . . , ∂pq (x) P of the elements of a minimal integrity basis (MIB) for the ring R[Rn ]G of G-invariant polynomi(p) is known to realize the orbit space Rn /G of G as a als. The domain of semi-positivity of P q semi-algebraic variety in the space R spanned by the variables p1 , . . . , pq . (p) can be obtained from the solutions of a universal differential equation (master The matrices P equation), which satisfy convenient initial conditions. The master equation and the initial conditions involve as free parameters only the degrees da of the pa (x)’s. This approach tries to bypass the actual impossibility of explicitly determining a set of basic polynomial invariants for each group. 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. Mathematics Subject Classifications (2000): 14L24, 13A50, 14L30. Key words: geometric invariant theory, linear group actions, orbit spaces, coregular algebraic linear groups.
1. Introduction Invariant functions under the transformations of a compact linear group (hereafter abbreviated in CLG) G, acting in an Euclidean space Rn , play an important role in physics, and the determination of their properties is often a basic problem to solve in many physical contexts, such as the determination of patterns of spontaneous symmetry breaking, the analysis of phase spaces and structural phase transitions (Landau’s theory), covariant bifurcation theory, crystal field theory and so on (see, for instance, [9, 6, 7, 21, 22, 5, 27] and references therein). A G-invariant function f (x), x ∈ Rn , takes on constant values along each orbit of G, thus, if one has to analyze its properties, it is certainly more economical, and generally more effective, to think of it as a function defined in the orbit space This paper is partially supported by INFN and MURST 40% and 60%.
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Rn /G of the action of G in Rn . In this way, it is possible to take fully into account the invariance properties of f (x), while maintaining its regularity properties, but avoiding the troubles that could be met, for instance in the determination of the minima, owing to their degeneracy along the G-orbits. This approach to the study of the properties of a G-invaria
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