  Let $B$ be the group of invertible upper-triangular complex $n\times n$
matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with
zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts
on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$,
$x\in\mathfrak{u}$.
  To each involution $\sigma$ in $S_n$, the symmetric group on $n$ letters, one
can assign the $B$-orbit $\Omega_{\sigma}\in\mathfrak{u}^*$. We present a
combinatorial description of the partial order on the set of involutions
induced by the orbit closures. The answer is given in terms of rook placements
and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$.
  Using results of F. Incitti, we also prove that this partial order coincides
with the restriction of the Bruhat--Chevalley order to the set of involutions.

Representation Theory;Combinatorics