Annals of Combinatorics Progress on Numerator Formulae Expansions for Afne Kac-Moody Algebras

AMS Subject Classication: 17B65 This paper is dedicated to the memory of Gian-Carlo Rota who, amongst other things in a lifetime of high academic achievement, did so much to enhance the study of combinatorics. He did this both directly through his own research and indirectly through those many colleagues to whom he acted as a source of inspiration and encouragement. Abstract. The Weyl-Kac character formula for afne Kac-Moody algebras is recast as a quo-tient whose numerator and denominator can both be expressed as innite sums of characters of irreducible highest weight representations of simple Lie subalgebra of the same rank. The denominator expansions, which coincide with well known Macdonald identities, are expressed here in terms of innite series of characters, specied by particular types of partitions, subject to rank-dependent modication rules. It is shown that certain numberings of the associated Young diagrams provide a convenient framework for writing down contributions to the corresponding numerator expansions. In the case of the seven innite series of afne Kac-Moody algebras that are indexed by their rank, progress is reported on the extent to which their numerator expansions can be completely determined