The character of the total power operation

TL;DR

This paper computes the total power operation in Morava $E$-theory for finite groups, providing explicit formulas and connecting it to classical operations and character maps, advancing understanding of algebraic topology and formal group laws.

Contribution

It introduces a formula for the total power operation in Morava $E$-theory using $GL_n(Q_p)$-actions and relates the character map to global power functors, offering new computational tools.

Findings

Explicit formula for total power operation in Morava $E$-theory.

Connection between character map and global power functors.

Specializations yield classical operations.

Abstract

In this paper we compute the total power operation for the Morava $E$-theory of any finite group up to torsion. Our formula is stated in terms of the $GL_n(Q_p)$-action on the Drinfeld ring of full level structures on the formal group associated to $E$-theory. It can be specialized to give explicit descriptions of many classical operations. Moreover, we show that the character map of Hopkins, Kuhn, and Ravenel from $E$-theory to $GL_n(Z_p)$-invariant generalized class functions is a natural transformation of global power functors on finite groups.