In the last decade many people have started to devise and implement algorithms related to Lie theory. The most noteworthy example of this is the LiE package in which most of the basic combinatorial aspects of Lie theory have been implemented, following the excellent description and tables in Bourbaki. There remain many, more complex aspects of Lie theory for which it would be useful to have a computer implementation of the structure. In [1] I lay the foundation for a computer algebra package for computations related to symmetric varieties. These varieties are defined as the spherical homogeneous spaces G/H with G a reductive algebraic group and H the fixed point group of an involution. They occur in many problems in representation theory, geometry, and number theory. Perhaps the best known application is in the representation theory of Lie groups. There the symmetric varieties are of fundamental importance in many problems, ranging from the representation theory on symmetric spaces to the characterization of the characters of the irreducible representations of a semisimple Lie group. In [2] I extended this algorithm to include real symmetric spaces as well. I hope to implement both these algorithms and develop such a computer algebra package over the next few years.

1. A. G. Helminck, Computing B-orbits on G/H, J. Symb. Comp. Vol. 21, (1996) 169-209.
2. A. G. Helminck, Computing orbits of minimal parabolic k-subgroups acting on symmetric k-varieties, J. Symb. Comp. To appear.