Some schemes related to the commuting variety

The commuting variety is the pairs of $n\times n$ matrices $(X,Y)$ such that $XY=YX$. We introduce the diagonal commutator scheme, $\big \{ (X,Y) : XY-YX$ is diagonal$\big \}$, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme $\big \{ (X,Y) : XY$ lower triangular, $YX$ upper triangular$\big \}$, which is again a reduced complete intersection, this time with $n!$ components (one for each permutation). The degrees of these components give interesting invariants of permutations.