Flat bases of invariant polynomials and (P)over-cap-matrices of E7 and E8

Let G be a compact group of linear transformations of a Euclidean space V. The G-invariant C-infinity functions can be expressed as C-infinity 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 semidefiniteness conditions of the (P) over cap -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. [Commun. Algebra 8, 373 (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 E-7 and E-8. In this paper the flat basic sets of invariant homogeneous polynomials of E-7 and E-8 and the corresponding (P) over cap -matrices are determined explicitly. Using the results here reported one is able to determine easily the (P) over cap -matrices corresponding to any other integrity basis of E-7 or E-8. From the (P) over cap -matrices one may then write down the equations and inequalities defining the orbit spaces of E-7 and E-8 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 E-7 and E-8 or one of the Lie groups E-7 and E-8 in their adjoint representations. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3272569]