Flat Bases of Invariant Polynomials and P-matrices of E7 and E8

الأصل · EN

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, 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 et al. 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.