Fundamental weights, permutation weights and Weyl character formula

Abstract

For a finite Lie algebra of rank N, the Weyl orbits of strictly dominant weights contain number of weights, where is the dimension of its Weyl group . For any , there is a very peculiar subset for which we always have For any dominant weight , the elements of are called permutation weights. It is shown that there is a one-to-one correspondence between the elements of and where is the Weyl vector of . 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 , calculation of the character for the irreducible representation will then be provided by 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, , which requires a sum over Weyl group elements, is studied explicitly. This will be instructive also for an explicit application of 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.