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

Abstract

Let $G$ be a compact group of linear transformations of an Euclidean space $V$. The $G$-invariant $C^\infty$ functions can be expressed as $C^\infty$ 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 {\em 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 $P$-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 $E_7$ or $E_8$. From the $P$-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.