Chains in weak order posets associated to involutions.

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of three different weak order posets: the set of all involutions in the symmetric group, the set of fixed point free involutions of the symmetric group, and the set of charged involutions. These distinguished sets of involutions parametrize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we characterize the maximal chains of an arbitrary lower order ideal in any of these three posets.