Linear representation theory of Klein four-group

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 $-1$ .