## Abstract

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, E_{6}, which requires a sum over 51 840 Weyl group elements, is studied explicitly. This will be instructive also for an explicit application of A_{5} 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.

Original language | English |
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Pages (from-to) | 1701-1707 |

Number of pages | 7 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 32 |

Issue number | 9 |

DOIs | |

Publication status | Published - 5 Mar 1999 |