Progress on Numerator Expansions for Affine Kac-Moody Algebras

The Weyl-Kac character formula for affine Kac-Moody algebras is recast as a quotient whose numerator and denominator can both be expressed as infinite 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 infinite series of characters, specified by particular types of partitions, subject to rank-dependent modification 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 infinite series of affine Kac-Moody algebras that are indexed by their rank, progress is reported on the extent to which their numerator expansions can be completely determined.