Indefinite Kac-Moody algebras of special linear type
The authors continue their investigations into the root multiplicities of certain classes of indefinite Kac-Moody algebras. [For the earlier papers in the series, see Adv. Math. 97, No. 2, 154-190 (1993; Zbl 0854.17026) and ibid. 105, 76-100 (1994; Zbl 0824.17025)]. In this paper, the authors study a class of algebras obtained from the special linear Lie algebra \(A_n\). This class includes the rank 2 hyperbolic Kac-Moody algebras and the algebras \(HA_1^{(1)}\), \(HA_2^{(2)}\), \(HG_2^{(1)}\) and \(HD_4^{(3)}\) previously studied by Kang and by Feingold and Frenkel. Refining the techniques of the earlier papers, the authors give three root multiplicity formulas. The first involves Littlewood-Richardson coefficients and Kostka numbers, but is applicable only to roots of small degree. The second and third are recursive and closed-form formulas giving the multiplicities of arbitrary roots as polynomials of Kostka numbers. The paper is very clearly written; the authors motivate and explain their results, and they provide a wealth of examples illustrating their various computational techniques.
scientific article; zbMATH DE number 910232