#### Volume 8, issue 1 (2008)

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Rings of symmetric functions as modules over the Steenrod algebra

### William M Singer

Algebraic & Geometric Topology 8 (2008) 541–562
##### Abstract

We write ${P}^{\otimes s}$ for the polynomial ring on $s$ letters over the field $ℤ∕2$, 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 $\mathsc{A}$. That is, we would like to determine the graded vector spaces $ℤ∕2{\otimes }_{\mathsc{A}}{\left({P}^{\otimes s}\right)}^{{\Sigma }_{s}}$. Our main result is stated in terms of a “bigraded Steenrod algebra” $\mathsc{ℋ}$. The generators of this algebra $\mathsc{ℋ}$, like the generators of the classical Steenrod algebra $\mathsc{A}$, 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 $ℤ∕2{\otimes }_{\mathsc{A}}{\left({P}^{\otimes s}\right)}^{{\Sigma }_{s}}$, 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 $\mathsc{ℋ}$.

##### Keywords
Steenrod algebra, cohomology of classifying spaces, cohomology of the Steenrod algebra, Adams spectral sequence, algebraic transfer, hit elements
##### Mathematical Subject Classification 2000
Primary: 13A50, 55S10
Secondary: 18G15, 55Q45, 55T15, 18G10