Symmetry classes of tensors and Semi-direct product of finite abelian groups

TL;DR

This paper establishes a necessary and sufficient condition for the existence of o*-bases in symmetry classes of tensors linked to semi-direct products of finite abelian groups, advancing understanding in tensor symmetry classifications.

Contribution

It provides the first general criterion for o*-basis existence in symmetry classes of tensors associated with semi-direct products of finite abelian groups.

Findings

Derived a necessary and sufficient condition for o*-basis existence

Applied the criterion to semi-direct products and wreath products

Enhanced the theoretical framework for tensor symmetry classes

Abstract

In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite groups yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.