Some implications of a conjecture of Zabrocki to the action of $S_{n}$ on polynomial differential forms

Abstract

The symmetric group acts on polynomial differential forms on $\mathbb{R}^{n}$ through its action by permuting the coordinates. In this paper the $S_{n}% $-invariants are shown to be freely generated by the elementary symmetric polynomials and their exterior derivatives. A basis of the alternants in the quotient of the ideal generated by the homogeneous invariants of positive degree is given. In addition, the highest bigraded degrees are given for the quotient. All of these results are consistent with predictions derived by Garsia and Romero from a recent conjecture of Zabrocki.