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

Abstract

Let n_n(C) be the algebra of strictly upper-triangular n x n matrices over the field of complex numbers and X_2 the subset of matrices of nilpotent order 2. Let B_n(C) be the group of invertible upper-triangular matrices acting on n_n(C) by conjugation. Let B(u) be the orbit of u in X_2 with respect to this action. Let S_n^2 be the subset of involutions in the symmetric group S_n. We define a new partial order on S_n^2 which gives the combinatorial description of the closure of B(u). We also construct an ideal I(B(u)) in symmetric algebra S(n_n(C)^* whose variety V(I(B(u))) equals the closure of B(u) (in Zariski topology). 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.