Hopf ring structure on the mod p cohomology of symmetric groups

Guerra, Lorenzo

doi:10.2140/agt.2017.17.957

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Hopf ring structure on the mod $p$ cohomology of symmetric groups

Lorenzo Guerra

Algebraic & Geometric Topology 17 (2017) 957–982

DOI: 10.2140/agt.2017.17.957

Correction: 10.2140/agt.2024.24.2385

Abstract

We describe a Hopf ring structure on $\oplus_{n \geq 0} H^{*} (Σ_{n}; ℤ_{p})$ , discovered by Strickland and Turner, where $Σ_{n}$ is the symmetric group of $n$ objects and $p$ is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For $p = 2$ this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.

Keywords

group cohomology, symmetric group, Hopf ring, Dyer–Lashof operations, Steenrod algebra, Mui invariants

Mathematical Subject Classification 2010

Primary: 20J06

References

Publication

Received: 9 November 2015

Revised: 16 September 2016

Accepted: 30 September 2016

Published: 14 March 2017

Correction: 16 July 2024

Authors

Lorenzo Guerra
Scuola Normale Superiore Piazza dei Cavalieri 7 I-56126 Pisa Italy

Volume 17, issue 2 (2017)