Compact Linear Group Research Articles

On the invariant theory and geometry of compact linear groups of cohomogeneity⩽3

The paper is concerned with compact linear groups of cohomogeneity ⩽ 3, namely closed subgroups G of the orthogonal group O( n) with dim R n ⧸G⩽ 3. The main issue is to determine the ring of invariants as well as the orbital distance metric of S n−1 ⧸ G, for all groups of cohomogeneity 3. The starting point is the known classification of compact connected linear groups of this type, and the basic idea is to fully utilize a technique usually referred to as the reduction principle. Although the calculations of principal isotropy groups and their normalizers can be quite technical, the approach is rather elementary. Some of the material is of an expository nature since we try to present complete results and also strive to be selfcontained wherever possible.

Universality in orbit spaces of compact linear groups

If {p 1(x),...,p q (x)} is a minimal integrity basis of the ideal of polynomial invariants of a compact coregular linear groupG, the orbit map , yields a diffeomorphic image of the orbit spaceR n /G. Starting from this fact, we point out some properties which are common to the orbit space of all the compact coregular linear groups of transformations ofR n . In particular we show that a contravariant metric matrix $$\hat P(p)$$ can be defined in the interior of , as a polynomial function of (p 1,...,p q ). We prove that the matrix $$\hat P(p)$$ , which characterizes the set , as it is positive semi-definite only forp∈ , can be determined as a solution of a canonical differential equation, which, for every compact coregular linear group, depends only on the numberq and on the degrees of the elements of the minimal integrity bases. This allows to determine all the isomorphism classes of the orbit spaces of the compact coregular linear groups through a determination of the equivalence classes of the corresponding matrices $$\hat P(p)$$ . Forq≦3 (orbit spaces with dimensions ≧3), the solutions $$\hat P(p)$$ of the canonical equation are explicitly determined and the number of their equivalence classes is shown to be finite. It is also shown that, with a convenient choice of the minimal integrity basis, the polynomial matrix elements of $$\hat P(p)$$ have only integer coefficients. Arguments are given in favour of the conjecture that our conclusions hold true for all values ofq. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking.

UNIVERSALITY IN ORBIT SPACES OF SYMMETRY GROUPS AND IN SPONTANEOUS SYMMETRY BREAKING

Universality properties of orbit spaces of compact linear groups are proved and related to universality properties of patterns of spontaneous symmetry breaking. In particular, it is argued that the orbit spaces of all the compact linear groups possessing a basis of fundamental polynomial invariants with the same degrees can be classified in a finite (and small for small dimensions and degrees) number of isomorphism classes.

On the 3j symmetries

The symmetry properties of the 3jm tensor for any finite or compact linear group are discussed using a wreath product construction. This is shown to provide a complete group theoretic explanation for all symmetry properties whether ‘‘essential’’ or ‘‘arbitrary.’’ The link with the similar—but distinct—method of inner plethysms is considered.

A theorem on orbit structures (strata) of compact linear Lie groups

We present a comprehensive constructive proof of a theorem characterizing the tangent space to a stratum (orbit structure) of the Euclidean space Rn, seat of an orthogonal representation of a compact group G. The characterization is made in terms of gradients of a complete set (integrity basis) of G-invariant polynomials. In a recent paper [M. Abud and G. Sartori, Phys. Lett. B 104, 147 (1981)], the theorem, which may be considered a generalization of a theorem by Michel [C. R. Acad. Sci. Ser. A 272, 433 (1971)], has been shown to be effective in the determination of the equations of the strata and in the determination of natural extrema of G-invariant functions.

The geometry of orbit-space and natural minima of Higgs potentials

Starting from an analysis of the geometry of orbit-space, we propose a new approach to the determination of those extrema of a G-invariant C ∞-function (G is a compact linear group), which are associated to an arbitrary, but fixed, residual symmetry group. When applied to Higgs potentials of unified gauge field theories, the method allows one to state conditions for the minima to exist and be natural (stable) and to clarify the origin of a class of pseudo-Goldstone bosons.

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR n , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR n . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.