Rings of symmetric functions as modules over the Steenrod algebra

Singer, William

doi:10.2140/agt.2008.8.541

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Rings of symmetric functions as modules over the Steenrod algebra

William M Singer

Algebraic & Geometric Topology 8 (2008) 541–562

DOI: 10.2140/agt.2008.8.541

Abstract

We write $P^{\otimes s}$ for the polynomial ring on $s$ letters over the field $ℤ ∕ 2$ , equipped with the standard action of $Σ_{s}$ , the symmetric group on $s$ letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring ${(P^{\otimes s})}^{Σ_{s}}$ as a module over the Steenrod algebra $A$ . That is, we would like to determine the graded vector spaces $ℤ ∕ 2 \otimes_{A} {(P^{\otimes s})}^{Σ_{s}}$ . 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 $S q^{0}$ is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces $ℤ ∕ 2 \otimes_{A} {(P^{\otimes s})}^{Σ_{s}}$ , for all $s \geq 0$ , into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over $ℋ$ .

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

References

Publication

Received: 25 October 2007

Accepted: 4 January 2008

Published: 12 May 2008

Authors

William M Singer
Department of Mathematics Fordham University Bronx, NY 10458 USA

Volume 8, issue 1 (2008)