Combinatorics of B-orbits and Bruhat-Chevalley order on involutions

Let G be the general linear group, B its standard Borel subgroup, the space of strictly upper-triangular matrices, its dual space, S (n) the symmetric group on n letters, the set of involutions of S (n) . To one can assign the B-orbit . Let sigma, . We compute dim Omega (sigma) and prove that Omega (tau) is contained in the Zariski closure of Omega (sigma) if and only if tau a parts per thousand currency sign sigma with respect to the Bruhat-Chevalley order. We also give a conjectural description of the closure of Omega (sigma) and discuss some connections with the geometry of Schubert varieties.