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

Understand

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.