Fundamental weights, permutation weights and Weyl character formula

Abstract

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(��^{++})$ of strictly dominant weights $��^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(��^{++})$, there is a very peculiar subset $\wp(��^{++})$ for which we always have $$ dim\wp(��^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ ��^+ $, the elements of $\wp(��^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(��^{++})$ and $\wp(��)$ where $��$ is the Weyl vector of $G_N$. The concept of signature factor which enters in 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 $��^+$, calculation of the character $ChR(��^+)$ for irreducible representation $R(��^+)$ will then be provided by $A_N$ multiplicity rules governing generalized Schur functions. The main idea is again to express everything in terms of the so-called {\bf Fundamental Weights} with which we obtain a quite relevant specialization in applications of Weyl character formula.6 pages, no figures, TeX, as will appear in Journal of Physics A:Mathematical and General