Editing Group representation (section)

== Examples ==
Consider the complex number ''u'' = e<sup>2πi / 3</sup> which has the property ''u''<sup>3</sup> = 1. The set ''C''<sub>3</sub> = {1, ''u'', ''u''<sup>2</sup>} forms a [[cyclic group]] under multiplication. This group has a representation ρ on <math>\mathbb{C}^2</math> given by:

:<math>
\rho \left( 1 \right) =
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
\qquad
\rho \left( u \right) =
\begin{bmatrix}
1 & 0 \\
0 & u \\
\end{bmatrix}
\qquad
\rho \left( u^2 \right) =
\begin{bmatrix}
1 & 0 \\
0 & u^2 \\
\end{bmatrix}.
</math>

This representation is faithful because ρ is a [[injective|one-to-one map]].

Another representation for ''C''<sub>3</sub> on <math>\mathbb{C}^2</math>, isomorphic to the previous one, is σ given by:

:<math>
\sigma \left( 1 \right) =
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
\qquad
\sigma \left( u \right) =
\begin{bmatrix}
u & 0 \\
0 & 1 \\
\end{bmatrix}
\qquad
\sigma \left( u^2 \right) =
\begin{bmatrix}
u^2 & 0 \\
0 & 1 \\
\end{bmatrix}.
</math>

The group ''C''<sub>3</sub> may also be faithfully represented on <math>\mathbb{R}^2</math> by τ given by:

:<math>
\tau \left( 1 \right) =
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
\qquad
\tau \left( u \right) =
\begin{bmatrix}
a & -b \\
b & a \\
\end{bmatrix}
\qquad
\tau \left( u^2 \right) =
\begin{bmatrix}
a & b \\
-b & a \\
\end{bmatrix}
</math>

where

:<math>a=\text{Re}(u)=-\tfrac{1}{2}, \qquad b=\text{Im}(u)=\tfrac{\sqrt{3}}{2}.</math>

A possible representation on <math>\mathbb{R}^3</math> is given by the set of cyclic permutation matrices ''v'':

:<math>
\upsilon \left( 1 \right) =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\qquad
\upsilon \left( u \right) =
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
\end{bmatrix}
\qquad
\upsilon \left( u^2 \right) =
\begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{bmatrix}
.</math>

Another example:

Let <math>V</math> be the space of homogeneous degree-3 polynomials over the complex numbers in variables <math>x_1, x_2, x_3. </math>

Then <math>S_3</math> acts on <math>V</math> by permutation of the three variables.

For instance, <math>(12)</math> sends <math>x_{1}^3</math> to <math>x_{2}^3</math>.