199 research outputs found
Mod-two cohomology of symmetric groups as a Hopf ring
We compute the mod-2 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 Stiefel-Whitney 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.Comment: 31 pages, 6 figure
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