<s>[INST] Irreducible representation of $S_3$ How can I show that this representation of $S_3$ is irreducible?
$$\rho\left(e\right)=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right),\,\,\rho\left(a_{1}\right)=\frac{1}{2}\left(\begin{array}{cc}
-1 & -\sqrt{3}\\
\sqrt{3} & -1
\end{array}\right),\,\,\rho\left(a_{2}\right)=\frac{1}{2}\left(\begin{array}{cc}
-1 & \sqrt{3}\\
-\sqrt{3} & -1
\end{array}\right),$$
 $$\rho\left(a_{3}\right)=\left(\begin{array}{cc}
-1 & 0\\
0 & 1
\end{array}\right),\,\,\rho\left(a_{4}\right)=\frac{1}{2}\left(\begin{array}{cc}
1 & \sqrt{3}\\
\sqrt{3} & -1
\end{array}\right),\,\,\rho\left(a_{5}\right)=\frac{1}{2}\left(\begin{array}{cc}
1 & -\sqrt{3}\\
-\sqrt{3} & -1
\end{array}\right).$$
 [/INST] Well, if it were reducible, $\rho(a_3)$ and $\rho(a_4)$ would share an eigenvector. Which is clearly not the case, because the standard basis is the only (up to scalar multiplication and permutation) basis of eigenvectors of $\rho(a_3)$.
</s>