# Adams operations and symmetries of representation categories

## TL;DR

This paper investigates how the monoidal structure and symmetries of representation categories of finite groups determine Adams operations, revealing their dependence on symmetry choices and classifying all such symmetries and autoequivalences.

## Contribution

It shows that the monoidal structure determines all odd Adams operations and classifies all symmetries and monoidal autoequivalences of finite group representation categories.

## Key findings

- Monoidal structure determines all odd Adams operations.
- Examples show monoidal equivalences may not preserve second Adams operations.
- Classification of all symmetries and autoequivalences of representation categories.

## 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 lambda-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.