Subgroups of $p$-divisible groups and centralizers in symmetric groups

Abstract:

We give a formula relating the transfer maps for the cohomology theories $E_{n}$ and $C_t$ to the transchromatic generalized character maps of a previous paper by the author. We then apply this to understand the effect of the transchromatic generalized character maps on Strickland’s isomorphism between the Morava $E$-theory of the symmetric group $\Sigma _{p^k}$ (modulo a transfer ideal) and the global sections of the scheme that classifies subgroups of order $p^k$ in the formal group associated to $E_{n}$. This provides an algebro-geometric interpretation to the $C_t$-cohomology of the class of groups arising as centralizers of finite sets of commuting elements in symmetric groups.