Group S3: Irreducible vs Reducible Representation

[LagrangeEuler]

Homework Statement

$e = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$,
$a =\frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}$.
$b =\frac{1}{2} \begin{bmatrix} 1 & \sqrt{3} \\[0.3em] \sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
$c= \begin{bmatrix} -1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$
$d=\frac{1}{2} \begin{bmatrix} -1 & \sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
$f=\frac{1}{2} \begin{bmatrix} -1 & -\sqrt{3} \\[0.3em] \sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
This is irreducible representation of group $S_3$. \\
Reducible representation of $S_3$ is
$e=d=f = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$
$a =b=c=\frac{1}{2} \begin{bmatrix} -1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & 1 \\[0.3em] \end{bmatrix}$
Why is better to use irreducible then reducible representation in this case and in general?

 

[Office_Shredder]

What do you mean by "better to use?" You haven't used any representations to do anything.

 

[LagrangeEuler]

In practice one always take some irreducible representation to work with. My question is why?