Description of B-orbit closures of order 2 in upper-triangular matrices

Let n_n(ℂ) be the algebra of strictly upper-triangular n×n matrices and let X₂ = {u ∈ n_n(ℂ) |u² = 0} be the subset of matrices of nilpotent order 2. Let B _n(ℂ) be the group of invertible upper-triangular matrices acting on n_n by conjugation. Let B_u be the orbit of u ∈ X₂ with respect to this action. Let S_n² be the subset of involutions in the symmetric group S_n We define a new partial order on S_n² which gives the combinatorial description of the closure of B_u We also construct an ideal I(B _u)⊂ S(n^*)whose variety V(I}(B_u) equals B_u. We apply these results to orbital varieties of nilpotent order 2 in sl_n(ℂ)in order to give a complete combinatorial description of the closure of such an orbital variety in terms of Young tableaux. We also construct the ideal of definition of such an orbital variety up to taking the radical.