On the image of the total power operation for Burnside rings

TL;DR

This paper studies the image of the total power operation in Burnside rings, constructing a subring with a universal property, and extends classical homomorphisms and counting lemmas to wreath products.

Contribution

It introduces a combinatorial subring capturing the image of power operations in Burnside rings and extends classical homomorphisms to wreath products.

Findings

Identifies a small subring containing the total power operation image

Constructs character maps and formulas for the total power operation

Generalizes Burnside's orbit counting lemma for wreath products

Abstract

We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wr\Sigma_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr \Sigma_n)$. As $n$ varies, the subrings $\mathring A(G,n)$ assemble into a commutative graded ring $\mathring A(G)$ with a universal property: $\mathring A(G)$ carries the universal family of power operations out of $A(G)$. We construct character maps for $\mathring A(G,n)$ and give a formula for the character of the total power operation. Using $\mathring A(G)$, we extend the Frobenius--Wielandt homomorphism of Dress--Siebeneicher--Yoshida to wreath products compatibly with the total power operation. Finally, we prove a generalization of Burnside's orbit counting lemma that describes the transfer map $A(G \wr \Sigma_n) \to A(\Sigma_n)$ on the subring $\mathring A(G,n)$.