Indefinite Kac-Moody algebras of special linear type

Abstract

From the special linear Lie algebra An = st(n + 1,Q we construct certain indefinite Kac-Moody Lie algebras IAn(a, b)and then use the representation theory of An to determine explicit closed form root multiplicity formulas for the roots a of /An(a, b) whose degree satisfies \deg(a)\ < 2a + 1. These expressions involve the well-known Littlewood-Richardson coefficients and Kostka numbers. Using the Euler-Poincare Principle and Kostant's formula, we derive two expressions, one of which is recursive and the other closed form, for the multiplicity of an arbitrary root a of IAn(a, b) as a polynomial in Kostka numbers.