Linear representation theory of cyclic group:Z3

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z3.
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Over the complex numbers

The cyclic group of order three, being an Abelian group, has the property that all its irreducible representations are one-dimensional, and all representations are thus completely reducible in terms of one-dimensional representations. There are three irreducible representations, the trivial representation, and two representations sending the generators to the cube roots of unity, in different orders.

Character table

Let $\omega$ be a primitive cube root of unity. In terms of the $i$-notation, $\omega$ can be taken as $e^{2\pi i/3}$ or $\cos (2\pi /3)+i\sin (2\pi /3)$, which is $(-1+i\sqrt{3})/2$. ${\omega }^2$ is the other primitive cube root of unity, and is given as $e^{-2\pi i/3}$ or $\cos (2\pi /3)-i\sin (2\pi /3)$ or $(-1-i\sqrt{3})/2$.

Representation/Conjugacy class	$e$ (identity element)	$x$ (generator)	$x^2$ (generator)
trivial representation	1	1	1
one nontrivial representation	1	$\omega$	${\omega }^2$
the other (conjugate) nontrivial representation	1	${\omega }^2$	$\omega$