Mod-two cohomology of symmetric groups as a Hopf ring

TL;DR

This paper computes the mod-2 cohomology of all symmetric groups as a Hopf ring, providing explicit algebraic and geometric descriptions, and explores related structures in invariant and representation theory.

Contribution

It offers a complete Hopf ring presentation, explicit cocycle representatives, and detailed structure over the Steenrod algebra for symmetric groups.

Findings

Explicit additive basis with graphical representation

Determined algebra structure over Steenrod algebra

Developed Hopf ring structures on symmetric invariants

Abstract

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.