## Algebraic & Geometric Topology

### Rings of symmetric functions as modules over the Steenrod algebra

William Singer

#### Abstract

We write $P⊗s$ for the polynomial ring on $s$ letters over the field $ℤ∕2$, equipped with the standard action of $Σs$, the symmetric group on $s$ letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring $(P⊗s)Σs$ as a module over the Steenrod algebra $A$. That is, we would like to determine the graded vector spaces $ℤ∕2⊗A(P⊗s)Σs$. Our main result is stated in terms of a “bigraded Steenrod algebra” $ℋ$. The generators of this algebra $ℋ$, like the generators of the classical Steenrod algebra $A$, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that $Sq0$ is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces $ℤ∕2⊗A(P⊗s)Σs$, for all $s≥0$, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over $ℋ$.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 541-562.

Dates
Accepted: 4 January 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796821

Digital Object Identifier
doi:10.2140/agt.2008.8.541

Mathematical Reviews number (MathSciNet)
MR2443237

Zentralblatt MATH identifier
1137.13005

#### Citation

Singer, William. Rings of symmetric functions as modules over the Steenrod algebra. Algebr. Geom. Topol. 8 (2008), no. 1, 541--562. doi:10.2140/agt.2008.8.541. https://projecteuclid.org/euclid.agt/1513796821

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