# Linear representation theory of Klein four-group

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The Klein four-group is an example of a rational representation group in the sense that all its representations can be realized over the field of rational numbers. In fact, all its representations can be realized over the two-element set $\pm 1$ and are one-dimensional, hence the representations described for characteristic zero generalize to any situation where the characteristic is not two.

We describe the Klein four-group as a four-element group with identity element $e$ and three non-identity elements $a,b,c$. Recall that each of the non-identity elements has order two and the product of any two distinct ones among them is the third one.

## Representations

There are four irreducible representations, all one-dimensional:

• The trivial representation: This sends all four elements to $1$.
• The representation with kernel $\{ e, a \}$: This sends $e$ and $a$ to $1$ and sends $b$ and $c$ to $-1$.
• The representation with kernel $\{ e, b \}$: This sends $e$ and $b$ to $1$ and sends $a$ and $c$ to $-1$.
• The representation with kernel $\{ e, c \}$: This sends $e$ and $c$ to $1$ and sends $a$ and $b$ to $-1$.

## Character table

Rep/element $e$ $a$ $b$ $c$
trivial 1 1 1 1
kernel $\{e ,a \}$ 1 1 -1 -1
kernel $\{e,b \}$ 1 -1 1 -1
kernel $\{e,c \}$ 1 -1 -1 1

## Realizability information

### Smallest ring of realization

Representation Smallest ring over which it is realized Smallest set of elements in matrix entries
trivial representation $\mathbb{Z}$ -- ring of integers $\{ 1 \}$
any of the nontrivial representations $\mathbb{Z}$ -- ring of integers $\{ 1, -1 \}$

### Smallest ring of realization as orthogonal matrices

Representation Smallest ring over which it is realized Smallest set of elements in matrix entries
trivial representation $\mathbb{Z}$ -- ring of integers $\{ 1 \}$
any of the nontrivial representations $\mathbb{Z}$ -- ring of integers $\{ 1, -1 \}$