<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 5 Issue 3" %>
Progress on Numerator Forulae Expansions for Affine Kac-Moody Algebras
Ronald C. King
Faculty of Mathematics Studies, University of Southampton, Southampton SO17 1BJ, England
Annals of Combinatorics 5 (3) p.381-395 September, 2001
AMS Subject Classification: 17B65
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.
Keywords: affine Kac-Moody algebras, Macdonald identities, character formulae, Young diagrams, affine Weyl groups, numerator expansions


1.  F. Aribaud, Une nouvelle démonstration d*un Théorème de R Bott and B Kostant, Bull. Soc. Math. France 95 (1967) 205每242.

2.  G.R.E. Black, R.C. King, and B.G. Wybourne, Kronecker products for semisimple Lie groups, J. Phys. A 16 (1983) 1555每1589.

3.  M.R. Bremmer, R.V. Moody, and J. Patera, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras, M Dekker, New York, 1985.

4.  E. Cartan, Sur la structure des groupes de transformations finis et continus, Thesis, Paris, 1894.

5.  J.F. Cornwell, Group Theory in Physics, Vol. III: Supersymmetries and Infinite-Dimensional Algebras, Academic Press, London, 1989.

6.  A. Hussin, Characters of affine Kac-Moody algebras, Ph.D. Thesis, University of Southampton, Southampton, 1995.

7.  A. Hussin and R.C. King, Affine Kac-Moody algebras and their representations, In: Symmetries and Structural Properties of Condensed Matter V, T. Lulek, W. Florek, and B. Lulek, Eds., World Scientific Press, Singapore, 1997, pp. 296每309.

8.  V.G. Kac, Infinite-dimensional Lie Algebras, 3rd Edition, Cambridge University Press, Cambridge, 1990.

9.  S.-J. Kang, Kac-Moody Lie algebras, spectral sequences, and theWitt formula, Trans. Amer. Math. Soc. 339 (1993) 463每495.

10.  R.C. King, S-functions and characters of Lie algebras and superalgebras, In: Invariant Theory and Tableaux, D. Stanton, Ed., Springer-Verlag, New York, 1990, pp. 226每261.

11.  B Kostant, Lie algebra cohomology and the generalized Borel-Weil Theorem, Ann. Math. 74 (1961) (2) 329每387.

12.  L. Liu, Kostant*s formula for Kac-Moody Lie algebras, J. Algebra 149 (1992) 155每178.

13.  I.G. Macdonald, Affine root systems and Dedekind*s h-function, Invent. Math. 15 (1972) 91每143.

14.  H. Weyl, Theorie der Darstellungen kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen III, Math. Zeit. 24 (1926) 377每396.