Permutation Weights for A_r^(1) Lie Algebras

When it is based on Kac-Peterson form of Affine Weyl Groups, Weyl-Kac character formula could be formulated in terms of Theta functions and a sum over finite Weyl groups. We, instead, give a reformulation in terms of Schur functions which are determined by the so-called Permutation Weights and there are only a finite number of permutation weights at each and every order of weight depths . Affine signatures are expressed in terms of an index which by definition is based on a decomposition of horizontal weights in terms of some Fundamental Weights.