Summing over the Weyl Groups of E_7 and E_8

Abstract

It is known that summations over Weyl groups of Lie algebras is a problem which enters in many areas of physics as well as in mathematics. For this, a method which we would like to call {\bf permutation weights} has been previously proposed for pairs $(G_N, A_{N-1})$ of Lie algebras. It is now extended for $(E_7, A_7)$ and also $(E_8, A_8)$. It is clear that these are the most non-trivial ones and hence deserve studying separately. In order to obtain the results of these summations in practice, it is shown that some simplifications occur in the method which is previously proposed for pairs $(A_N, A_{N-1})$ in an unpublished work.