David Mehrle
University of Kentucky
Title
The image of power operations on Burnside rings
Abstract
The Burnside ring $A(G)$ of a finite group $G$ is the archetypal ring in $G$-equivariant homotopy theory, appearing as the zeroth stable homotopy of the $G$-equivariant sphere spectrum. As part of the structure of this homotopy data, Burnside rings admit power operations sending a finite $G$-set $X$ to the set $X^n$ with compatible actions of both $G$ and a symmetric group $\Sigma_n$. We study the smallest subring of $A(G\wr \Sigma_n)$ generated by a subset of the canonical basis and containing the image of these power operations. We provide three equivalent characterizations of this ring: (1) by universal property; (2) by generators and relations; and (3) by means of a character theory / table of marks. Using this character theory, we give formulas for the power operations in the table of marks.
