Linear representation theory of Klein four-group

This article gives specific information, namely, linear representation theory, about a particular group, namely: 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	$Z$ -- ring of integers	${1}$
any of the nontrivial representations	$Z$ -- 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	$Z$ -- ring of integers	${1}$
any of the nontrivial representations	$Z$ -- ring of integers	${1, - 1}$