Computing representatives of nilpotent orbits of θ-groups

Two algorithms are described for finding representatives of the nilpotent orbits of a θ-group, corresponding to a Z/mZ-grading of a simple Lie algebra g over C. The first algorithm uses the classification of the nilpotent orbits in g, an idea also used in Đoković (1988a). To get a working algorithm from it, we combine this idea with a new method for computing normal sl2-triples. The second algorithm is based on Vinberg’s theory of carrier algebras, that reduces the classification of nilpotent orbits to the classification of subalgebras of g with certain properties. We describe an algorithm for the latter problem, using a method for classifying π-systems.The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We briefly comment on their performance. At the end of the paper the algorithms are used to study the nilpotent orbits of θ-groups, where θ is an N-regular automorphism of a simple Lie algebra of exceptional type.