Modtwo cohomology of symmetric groups as a Hopf ring
Abstract
We compute the mod2 cohomology of the collection of all symmetric groups as a Hopf ring, where the second product is the transfer product of Strickland and Turner. We first give examples of related Hopf rings from invariant theory and representation theory. In addition to a Hopf ring presentation, we give geometric cocycle representatives and explicitly determine the structure as an algebra over the Steenrod algebra. All calculations are explicit, with an additive basis which has a clean graphical representation. We also briefly develop related Hopf ring structures on rings of symmetric invariants and end with a generating set consisting of StiefelWhitney classes of regular representations v2. Added new results on varieties which represent the cocycles, a graphical representation of the additive basis, and on the Steenrod algebra action. v3. Included a full treatment of invariant theoretic Hopf rings, refined the definition of representing varieties, and corrected and clarified references.
 Publication:

arXiv eprints
 Pub Date:
 September 2009
 arXiv:
 arXiv:0909.3292
 Bibcode:
 2009arXiv0909.3292G
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Rings and Algebras;
 20J06;
 20B30
 EPrint:
 31 pages, 6 figures