Representations of The miraculous Klein group

TL;DR

This paper classifies all representations of the Klein group over a field of two elements, using visual methods and exploring applications to duality and Heller shifts in representation theory.

Contribution

It provides a complete description of Klein group representations over GF(2), employing visual techniques and connecting to advanced concepts like duality and Heller shifts.

Findings

Classified all Klein group representations over GF(2).

Introduced visual methods for understanding these representations.

Explored implications for duality and Heller shifts.

Abstract

The Klein group contains only four elements. Nevertheless this little group contains a number of remarkable entry points to current highways of modern representation theory of groups. In this paper, we shall describe all possible ways in which the Klein group can act on vector spaces over a field of two elements. These are called representations of the Klein group. This description involves some powerful visual methods of representation theory which builds on the work of generations of mathematicians starting roughly with the work of K. Weiestrass. We also discuss some applications to properties of duality and Heller shifts of the representations of the Klein group.