Algorithms for Computing Characters for Symmetric Spaces

The classification theorem for semisimple Lie algebras states that up to isomorphy any finite dimensional semisimple Lie algebra g defined over an algebraically closed field is uniquely determined by the root system of a maximal toral subalgebra. Moreover, if g is simple, then this root system is irreducible and its Dynkin diagram must be one of An, Bn, Cn, Dn, E6, E7, E8, F4, or G2. This is fundamental to the study of finite-dimensional semisimple Lie algebras over algebraically closed fields. A. G. Helminck established an analogous result for local symmetric spaces where he identified twentyfour graphical structures called involution or θ-diagrams (see [Hel88]). Implicit in each of these diagrams are two root systems 8(a) and 8(t) with a a maximal toral subalgebra in a local symmetric space p and t ⊃ a a maximal toral subalgebra in the corresponding semisimple Lie algebra g. The root system 8(a) can be described as the image of 8(t) under a projection π derived from an involution θ on 8(t). The weight lattices associated with 8(t) and 8(a) are denoted by 3t and 3a, respectively. In [GH06] it was shown that if π is extended linearly from 8(t) to 3t then π(3t )=3a. Explicit algorithmic formulations for the characters of each of these lattices (in terms of the other) was also given. In this paper we provide a methodology for implementing these algorithms for use in a computer algebra package.