The transfer in mod-p group cohomology between \Sigma_p \int \Sigma_{p^{n-1}}, \Sigma_{p^{n-1}} \int \Sigma_p and \Sigma_{p^n}

TL;DR

This paper computes transfer maps in mod-p cohomology for symmetric groups and their subgroups, providing explicit invariants, bases for invariant rings, and relations to Dickson algebra generators.

Contribution

It introduces explicit calculations of transfer maps and invariant rings in mod-p cohomology for symmetric groups and their subgroups, linking to Dickson algebra structures.

Findings

Explicit transfer maps in mod-p cohomology are computed.

A free module basis for rings of invariants over Dickson algebra is established.

The transfer map coincides with a natural epimorphism in a specific ideal.

Abstract

In this work we compute the induced transfer map: $$\bar\tau^\ast: \func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast: H^\ast (\Sigma_{p^n}) \to H^\ast(V))$$ in $\func{mod}p$-cohomology. Here $\Sigma_{p^{n}}$ is the symmetric group acting on an $n$-dimensional $\mathbb F_p$ vector space $V$, $G=\Sigma_{p^{n},p}$ a $p$-Sylow subgroup, $\Sigma_{p^{n-1}}\int \Sigma_{p}$, or $\Sigma_{p}\int \Sigma_{p^{n-1}}$. Some answers are given by natural invariants which are related to certain parabolic subgroups. We also compute a free module basis for certain rings of invariants over the classical Dickson algebra. This provides a computation of the image of the appropriate restriction map. Finally, if $ \xi :\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast}: H^\ast(\Sigma_{p^n}) \to H^\ast(V)) $ is the natural epimorphism, then we prove that $\bar\tau^\ast=\xi$ in the ideal generated by the top Dickson algebra generator.