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arXiv · 2405.06661

On the image of the total power operation for Burnside rings

Nathan Cornelius·Lewis Dominguez·David Mehrle·Lakshay Modi·Millie Rose·Nathaniel Stapleton
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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 $\mathring A(G,n)$.

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Nathan Cornelius, Lewis Dominguez, David Mehrle, Lakshay Modi, Millie Rose, Nathaniel Stapleton. 2024-04-24. On the image of the total power operation for Burnside rings. https://arxiv.org/abs/2405.06661

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