  Let A be a local commutative principal ideal ring. We study the double coset
space of GL(n,A) with respect to the subgroup of upper triangular matrices.
Geometrically, these cosets describe the relative position of two full flags of
free primitive submodules of A^n. If k is the length of the ring, we determine
for which of the pairs (n,k) the double coset space depend on the ring in
question. For n=3, we give a complete parametrisation of the double coset space
and provide estimates on the rate of growth of the number of double cosets.
