Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 1 (2008), 541-562.
Rings of symmetric functions as modules over the Steenrod algebra
We write for the polynomial ring on letters over the field , equipped with the standard action of , the symmetric group on letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring as a module over the Steenrod algebra . That is, we would like to determine the graded vector spaces . 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 , satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces , for all , into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over .
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 541-562.
Received: 25 October 2007
Accepted: 4 January 2008
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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