Cohomology of symmetric and alternating groups

The homology of symmetric groups has been well-known since Nakaoka’s seminal work in 1962 and then Cohen-Lada-May’s reformulation and extension in 1974.  But the cup co-product structure of the latter relies on Adem relations, and so has been of limited value in applications, for example providing no insight as to the number of ring generators or existence of nilpotent elements.  For that reason, mathematicians such Madsen, Milgram, Magannis, Adem and ultimately Feshbach devoted considerable energy to understanding cup product structure in more detail, building on its beautiful connection to invariant theory.

What was missing, from our perspective, was an induction or transfer product which along with known structures forms a Hopf ring, first discovered by Strickland and Turner in 1997. Using this, Giusti, Salvatore and myself have a “one-line” description of the mod-two cohomology of symmetric groups published in 2012, and Giusti and I have just finished a corresponding presentation for alternating groups, which is substantially more technically challenging.   Such structure occurs for representations of symmetric or alternating groups, including at finite characteristic, and would be interesting to utilize in that setting.   

In this talk, I will give a hands-on introduction to our calculation for symmetric groups and then talk about the broader picture before discussing alternating groups.