On the Calculation of Group Characters

It is known that characters of irreducible representations of finite Lie algebras can be obtained using theWeyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras of lower rank. By starting from the Weyl character formula, we show that these characters can be re-expressed without referring to Weyl group summations. Some useful technical points are given in detail for the instructive example of G2 Lie algebra.Comment: 6 pages, no figure, Plain Te

On Poincare Polynomials of Hyperbolic Lie Algebras

We have general frameworks to obtain Poincare polynomials for Finite and also Affine types of Kac-Moody Lie algebras. Very little is known however beyond Affine ones, though we have a constructive theorem which can be applied both for finite and infinite cases. One can conclusively said that theorem gives the Poincare polynomial P(G) of a Kac-Moody Lie algebra G in the product form P(G)=P(g) R where g is a precisely chosen sub-algebra of G and R is a rational function. Not in the way which theorem says but, at least for 48 hyperbolic Lie algebras considered in this work, we have shown that there is another way of choosing a sub-algebra in such a way that R appears to be the inverse of a finite polynomial. It is clear that a rational function or its inverse can not be expressed in the form of a finite polynomial. Our method is based on numerical calculations and results are given for each and every one of 48 Hyperbolic Lie algebras. In an illustrative example however, we will give how above-mentioned theorem gives us rational functions in which case we find a finite polynomial for which theorem fails to obtain.Comment: 14 pages, 7 Figures, Plain Te