Linear representation theory of cyclic group:Z3

This article is about the linear representation theory of the following group: cyclic group of order three

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 cuberoots of unity, in different orders.

Character table

Let $\omega$ be a primitive cuberoot of unity.

Representation/Conjugacy class	$e$ (identity element)	$x$ (generator)	$x^2$ (generator)
trivial representation	1	1	1
--	1	$\omega$	${\omega }^2$
--	1	${\omega }^2$	$\omega$