Invariant theory to study symmetry breakings in the adjoint representation of E8

Abstract

To study analytically the possible symmetry breakings that may take place in the adjoint representation of the compact Lie group E8 one may take advantage of the knowledge of the basic polynomials of its Weyl group, that is of the finite reflection group E8, and of the (polynomial) equations and inequalities defining its orbit space. This orbit space is stratified and each of its strata represents a possible symmetry breaking, both for the adjoint representation of the Lie group E8 and for the defining representation of the finite reflection group E8. The concrete determination of the equations and inequalities defining the strata is not yet done but it is possible in principle using rank and positive semi-definiteness conditions on a symmetric 8 × 8 matrix whose matrix elements are real polynomials in 8 variables. This matrix, called the -matrix, has already been determined completely. In this article it will be reviewed how one may determine the equations and inequalities defining the strata of the orbit space and how these equations and inequalities may help to study spontaneous symmetry breakings. Here the focus is on E8, because some of the results for E8 are new, but all what it is here said for E8 may be repeated for any other compact simple Lie group. The -matrices of all irreducible reflection groups have in fact already been determined.