Fundamental Weights, Permutation Weights and Weyl Character Formula

Abstract

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(\Lambda^{++})$ of strictly dominant weights $\Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(\Lambda^{++})$, there is a very peculiar subset $\wp(\Lambda^{++})$ for which we always have $$ dim\wp(\Lambda^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ \Lambda^+ $, the elements of $\wp(\Lambda^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(\Lambda^{++})$ and $\wp(\rho)$ where $\rho$ 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 $\Lambda^+$, calculation of the character $ChR(\Lambda^+)$ for irreducible representation $R(\Lambda^+)$ 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.