Abstract
Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual λ-ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, determines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and to show that monoidal equivalences that preserve the second Adams operations do not have to be symmetric. Along the way, we classify all possible symmetries and all monoidal autoequivalences of representation categories of finite groups.
| Original language | English |
|---|---|
| Pages (from-to) | 501-523 |
| Number of pages | 23 |
| Journal | Indiana University Mathematics Journal |
| Volume | 70 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2021 |
Bibliographical note
Acknowledgements. Both authors received support from the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The first author was also supported by the Research Training Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory.” Thanks are due to Alexei Davydov, Lars Hesselholt, Victor Ostrik, and Björn Schuster for discussions, and the referee for the valuable comments on the exposition.Keywords
- Adams operations
- Representation rings
- Symmetric monoidal categories
- λ-rings