On the image of the total power operation for Burnside rings
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913350131122176 |
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| author | Cornelius, Nathan Dominguez, Lewis Mehrle, David Modi, Lakshay Rose, Millie Stapleton, Nathaniel |
| author_facet | Cornelius, Nathan Dominguez, Lewis Mehrle, David Modi, Lakshay Rose, Millie Stapleton, Nathaniel |
| contents | We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wrΣ_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr Σ_n)$. 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 $A(G \wr Σ_n) \to A(Σ_n)$ on the subring $\mathring A(G,n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the image of the total power operation for Burnside rings Cornelius, Nathan Dominguez, Lewis Mehrle, David Modi, Lakshay Rose, Millie Stapleton, Nathaniel Rings and Algebras Algebraic Topology Group Theory We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wrΣ_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr Σ_n)$. 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 $A(G \wr Σ_n) \to A(Σ_n)$ on the subring $\mathring A(G,n)$. |
| title | On the image of the total power operation for Burnside rings |
| topic | Rings and Algebras Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2405.06661 |