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

TL;DR

This paper develops new algebraic and combinatorial methods to decompose Kronecker powers of harmonics related to symmetric groups, providing formulas and generalized statistics for polynomial ring actions and evaluations.

Contribution

It introduces a novel decomposition approach for symmetric group actions on polynomial rings and generalizes the comaj statistic for permutations and tableaux.

Findings

Formulas for restrictions from GL_n to S_n in Schur-Weyl duality.

A generalized comaj statistic encompassing standard tableaux.

A generalized principal evaluation for Schur functions and quasisymmetric functions.

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.