Fundamental weights, permutation weights and Weyl character formula

For a finite Lie algebra G_N of rank N, the Weyl orbits W(Λ⁺⁺) of strictly dominant weights Λ⁺⁺ contain dim W(G_N) number of weights, where dim W(G_N) is the dimension of its Weyl group W(G_N). For any W(Λ⁺⁺), there is a very peculiar subset ρ(Λ⁺⁺) for which we always have dim ρ(Λ⁺⁺) = dim W(G_N)/dim W(A_N-1). For any dominant weight Λ⁺, the elements of ρ(Λ⁺) are called permutation weights. It is shown that there is a one-to-one correspondence between the elements of ρ(Λ⁺⁺) and ρ(ρ) where ρ is the Weyl vector of G_N. The concept of the signature factor which enters the Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant Λ⁺, calculation of the character Ch R(Λ⁺) for the irreducible representation R(Λ⁺) will then be provided by A_N multiplicity rules governing the generalized Schur functions The main idea is again to express everything in terms of the so-called fundamental weights with which we obtain a quite relevant specialization in applications of the Weyl character formula. To provide simplifications in practical calculations, a reduction formula governing the classical Schur functions is also given. As the most suitable example, E₆, which requires a sum over 51 840 Weyl group elements, is studied explicitly. This will be instructive also for an explicit application of A₅ multiplicity rules. As a result, it will be seen that the Weyl or Weyl-Kac character formulae find explicit applications no matter how large the rank of the underlying algebra.