The ladder game

The ladder game is an algorithm for the stepwise determination of transversals
of the sets of double cosets to for a subgroup of a group and a sequence of subgroups, a
socalled ladder, of with either or for all .
We call a step from to with a step down. A double coset of and splits into double cosets of and :
for a
transversal of . Therefore, a transversal of the double cosets is a subset of where means a transversal of the double
cosets. This subset has to be determined.
A step up from to is a step with . Such double cosets of and fuse to a double coset of and :
for a
transversal of . We obtain a transversal of the double cosets of and as a subset of a transversal of the
double cosets of and .
To keep down the complexity
of this algorithm, we should try to choose a ladder with a small index between and for all , because during each step, dow or up,
respectively, we run through a transversal of and , respectively.
In order to visualize the
ladder game we introduce a graph which we call orbit graph:
 The double cosets, , correspond to the vertices of the graph.
 Two vertices are connected by an edge iff
the corresponding double cosets and fulfill
 or .
 Each vertex is labeled by a tuple , where means the step in the ladder
game, is the th double coset of and and is the order of
the stabilizer of the corresponding double coset in .
Let us
return to the determination of transversals of and of We need a ladder of containing
and . For this purpose we take the following sequence of parabolic subgroups
(generalizing this way the ladder game that we applied to designs on sets,
where we took a ladder of Young subgroups!)
$$
$$
$$
$$
$$
$$
$$
as a
ladder. Starting from the trivial transversal of we can determine step by step all transversals
of for all .
Let us display a concrete
example: We take the parameters and, for we
choose , the complete monomial group, which is isomorph to the wreath product . Then we get the following orbit graph:
In this
graph the vertices with the labels correspond to the orbits of on the
set of supspaces
of and the vertices labeled by the tuples correspond to the orbits of on the
set of subspaces.