Flat bases of invariant polynomials and P̂-matrices of E7 and E8

Abstract

Let G be a compact group of linear transformations of a 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 semidefiniteness conditions of the P̂-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 E7 and E8. In this paper the flat basic sets of invariant homogeneous polynomials of E7 and E8 and the corresponding P̂-matrices are determined explicitly. Using the results here reported one is able to determine easily the P̂-matrices corresponding to any other integrity basis of E7 or E8. From the P̂-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.