---
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.'
address:
- |
    Department of Mathematics\
    Illinois State University\
    Normal, IL 61761 USA
- |
    Department of Mathematics\
    University of Western Ontario\
    London, ON, N6A 5B7, Canada
author:
- Sunil Chebolu
- Ján Minác
bibliography:
- 'lit.bib'
title: Representations of The miraculous Klein group
---

Introduction
============

Consider the familiar complex plane $\mathbb{C} = \{ x+ iy \, | \, x, y \text{ are real numbers} \}$ with two reflections $\sigma$ and $\tau$ in the standard axes $X$ and $Y$ respectively. Precisely, we have $$\begin{aligned}
\sigma(x+iy) & =  & x - iy, 