Summing over the Weyl Groups of E_7 and E_8

TL;DR

This paper extends a permutation weight method to compute summations over the Weyl groups of the complex Lie algebras E_7 and E_8, which are highly non-trivial cases, and discusses simplifications for practical calculations.

Contribution

The paper generalizes a previously proposed permutation weight method to the complex Lie algebras E_7 and E_8, enabling new calculations of Weyl group summations.

Findings

Method successfully extended to E_7 and E_8

Simplifications improve computational practicality

Addresses complex non-trivial Lie algebra cases

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it is shown that some simplifications occur in the method which is previously proposed for pairs $(A_N, A_{N-1})$ in an unpublished work.