The mod‐2 cohomology rings of symmetric groups

We present a new additive basis for the mod‐2 cohomology of symmetric groups, along with explicit rules for multiplication and application of Steenrod operations in that basis. The key organizational tool is a Hopf ring structure introduced by Strickland and Turner. We elucidate some of the relationships between our approach and previous approaches to the homology and cohomology of symmetric groups.