Rings of symmetric functions as modules over the Steenrod algebra

Abstract

We write $P^{\otimes s}$ for the polynomial ring on $s$ letters over the field $\mathbb{Z}∕2$, equipped with the standard action of ${\Sigma }_s$, the symmetric group on $s$ letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring ${\left(P^{\otimes s}\right)}^{{\Sigma }_s}$ as a module over the Steenrod algebra $\mathcal{A}$. That is, we would like to determine the graded vector spaces $\mathbb{Z}∕2{\otimes }_{\mathcal{A}}{\left(P^{\otimes s}\right)}^{{\Sigma }_s}$. Our main result is stated in terms of a “bigraded Steenrod algebra” $\mathcal{\mathscr{H}}$. The generators of this algebra $\mathcal{\mathscr{H}}$, like the generators of the classical Steenrod algebra $\mathcal{A}$, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that $\mathbb{S}{q}^0$ is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces $\mathbb{Z}∕2{\otimes }_{\mathcal{A}}{\left(P^{\otimes s}\right)}^{{\Sigma }_s}$, for all $s\ge 0$, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over $\mathcal{\mathscr{H}}$.