A remark on automorphisms of tensor spaces

TL;DR

This paper explores the automorphism groups of tensor spaces, demonstrating that many permutation groups can serve as automorphism groups and that tensor spaces can exhibit unusual behaviors as representations.

Contribution

It shows that a wide range of permutation groups can be realized as automorphism groups of tensor spaces, revealing complex behaviors in their representations.

Findings

Permutation groups can be automorphism groups of tensor spaces

Tensor spaces can have pathological behaviors as representations

A large class of permutation groups are realizable as automorphism groups

Abstract

A tensor space is a vector space equipped with a finite collection of multi-linear forms. In recent years, a rich theory of infinite dimensional tensor spaces has emerged. In this note, we show that a large class of permutation groups can occur as the automorphism groups of such tensor spaces. Using this, we show that a tensor space can behave somewhat pathologically as a representation of its automorphism group.