Kronecker powers of harmonics, polynomial rings, and generalized principal evaluations

Abstract

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.

Stability theorems for multiplicities in graded $S_n$-modules

“…(Here if µ = (µ 1 ) then we take µ 2 = 0.) One key aspect of the proof of this stabilization result is the following result of the first named author of this paper (see [7]). The total degree version found in Proposition 5.2 of [10] can also be used to provide an alternate proof.…”