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

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 2 A(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 2 A(Ps)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 .

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
Received: 25 October 2007
Accepted: 4 January 2008
Published: 12 May 2008
William M Singer
Department of Mathematics
Fordham University
Bronx, NY 10458