If {p_1(x), ..., p_q(x)} is a minimal integrity basis of the ideal of polynomial invariants of a compact coregular linear group G, the orbit map p=(p_1(x) .... ,p_q(x)):R^n->R^q,\ud yields a diffeomorphic image S = p(R^n) \subset R^q of the orbit space R^n/G. Starting from this fact, we point out some properties which are common to the orbit spaces of all the compact coregular linear groups of transformations of R^n. In particular we show that a contravariant metric matrix P(p) can be defined in the interior of S, as\ud a polynomial function of (p_1, ...,p_q). We prove that the matrix P(p), which characterizes the set S, as it is positive semi-definite only for p \in S, can be determined as a solution of a canonical differential equation, which, for every compact coregular linear group, depends only on the number q and on the degrees of the elements of the minimal integrity bases. This allows to determine all the\ud isomorphism classes of the orbit spaces of the compact coregular linear groups through a determination of the equivalence classes of the corresponding matrices\ud P(p). For q<3 (orbit spaces with dimensions < 3), the solutions P(p) of the canonical equation are explicitly determined and the number of their equivalence\ud 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 P(p) have only integer\ud coefficients. Arguments are given in favour of the conjecture that our conclusions hold true for all values of q. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking
The evolution of spatially homogeneous and isotropic cosmological models containing a perfect fluid with equation of state p = wρ and a cosmological constant Λ is investigated for arbitrary combinations of w and Λ, using standard qualitative analysis borrowed from classical mechanics. This approach allows one to consider a large variety of situations, appreciating similarities and differences between models, without solving the Friedmann equation, and is suitable for an elementary course in cosmology.
Let G be a compact group of linear transformations of an Euclidean space V . The G-invariant C ∞ functions can be expressed as C ∞ functions of a finite basic set of G-invariant homogeneous polynomials, sometimes called an integrity basis. The mathematical description of the orbit space V /G depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semi-definiteness conditions of the P -matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of G-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If G is an irreducible finite reflection group, Saito, Yano and Sekiguchi in 1980 characterized some special basic sets of G-invariant homogeneous polynomials that they called flat. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups E7 and E8. In this paper the flat basic sets of invariant homogeneous polynomials of E7 and E8 and the corresponding P -matrices are determined explicitly. Using the results here reported one is able to determine easily the P -matrices corresponding to any other integrity basis of E7 or E8. From the P -matrices one may then write down the equations and inequalities defining the orbit spaces of E7 and E8 relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups E7 and E8 or one of the Lie groups E7 and E8 in their adjoint representations.
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