  The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX.
We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal},
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 {(X,Y) :
XY lower triangular, YX upper triangular}, 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.
