In this paper we present an algorithm to compute the orbits uf a minimal pa
rabolic k-subgroup acting on a symmetric k-variety and most of the combinat
orial structure of the orbit decomposition. This algorithm can Le implement
ed in LiE, GAP4, Magma, Maple or in a separate program. These orbits are es
sential in the study of symmetric k-varieties and their representations. In
a similar way to the special case of a Borel subgroup acting on the symmet
ric variety, (see A. G. Helminck. Computing B-orbits on G/H. J. Symb. Compu
t., 21, 169-209, 1996.) one can use the associated twisted involutions in t
he restricted Weyl group to describe these orbits (sce A. G. Helminck and S
. P. Wang, On rationality properties of involutions of reductive groups. Ad
v. Math., 99, 26-96, 1993). However, the orbit structure in this case is mu
ch more complicated than the special case of orbits of a Borel subgroup. We
will first modify the characterization of the orbits of minimal parabolic
K-subgroups acting on the symmetric k-varieties given in Helminck and Wang
(1993), to illuminate the similarity to the one for orbits: of a Borel subg
roup acting on a symmetric variety in Helminck (1996). Using this character
ization we show how the algorithm in Helminck (1996) can be adjusted and ex
tended to compute these twisted involutions as well. (C) 2000 Academic Pres
s.