Algebraic & Geometric Topology

Rings of symmetric functions as modules over the Steenrod algebra

William Singer

Full-text: Open access

Abstract

We write Ps for the polynomial ring on s letters over the field 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 (Ps)Σs as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces 2A(Ps)Σs. Our main result is stated in terms of a “bigraded Steenrod algebra” . The generators of this algebra , 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 2A(Ps)Σs, for all s0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over .

Article information

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

Dates
Received: 25 October 2007
Accepted: 4 January 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 55S10: Steenrod algebra
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula [See also 55U25] 55Q45: Stable homotopy of spheres 55T15: Adams spectral sequences 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25]

Keywords
Steenrod algebra cohomology of classifying spaces cohomology of the Steenrod algebra Adams spectral sequence algebraic transfer hit elements

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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