Numerator formulae expansions for affine Kac-Moody algebras

Ronald C. King

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


The character of each integrable highest weight irreducible representation of an affine Kac-Moody algebra is expressed as the ratio of a numerator and a denominator, each of which may itself be expressed as an infinite sum of characters of irreducible representations of a maximal simple Lie subalgebra of the affine Kac-Moody algebra. The denominator expansions are nothing other than the celebrated Macdonald identities. In the case of the seven infinite series of affine Kac-Moody algebras that are indexed by their rank, each denominator expansion may be rendered in a rank-independent way. This had previously been carried out through the use of Schur function methods, but it is shown that the same results may be arrived at by making use of the coset representatives of the relevant affine Weyl group with respect to a finite Weyl subgroup. The use of these same coset representatives allows one to construct new numerator expansions. The expansions themselves are found to have a combinatorial structure involving a remarkable interplay between the Kac-Dynkin basis involving fundamental weights and the Euclidean basis related to Young diagrams and partitions. A procedure for determining numerator expansions is described and exemplified.

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