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2008 Rings of symmetric functions as modules over the Steenrod algebra
William Singer
Algebr. Geom. Topol. 8(1): 541-562 (2008). DOI: 10.2140/agt.2008.8.541


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 .


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William Singer. "Rings of symmetric functions as modules over the Steenrod algebra." Algebr. Geom. Topol. 8 (1) 541 - 562, 2008.


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

MathSciNet: MR2443237
zbMATH: 1137.13005
Digital Object Identifier: 10.2140/agt.2008.8.541

Primary: 13A50, 55S10
Secondary: 18G10, 18G15, 55Q45, 55T15

Rights: Copyright © 2008 Mathematical Sciences Publishers


Vol.8 • No. 1 • 2008
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