  Our main goal is to compute the decomposition of arbitrary Kronecker powers
of the Harmonics of $S_n$. To do this, we give a new way of decomposing the
character for the action of $S_n$ on polynomial rings with $k$ sets of $n$
variables. There are two aspects to this decomposition. The first is algebraic,
in which formulas can be given for certain restrictions from $GL_n$ to $S_n$
occurring in Schur-Weyl duality. The second is combinatorial. We give a
generalization of the $comaj$ statistic on permutations which includes the
$comaj$ statistic on standard tableaux. This statistic allows us to write a
generalized principal evaluation for Schur functions and Gessel Fundamental
quasisymmetric functions.
