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 $$-1$$.