I am trying to compute the Fourier transformation of the symmetric group $S_3$ following the section 4 of Quantum Computing and the Hunt for Hidden Symmetry.
The multiplication table of $S_3$ is as follows.
At this point I am supposed to come up with a representation $\rho : S_3 \to GL(V)$ where $GL(V)$ is the collection of invertible linear operators on a $\mathbb{C}$ vector space.
So, I choose $6$ $3 \times 3$ permutation matrices each corresponding to the permutation element because they are invertible.
All these matrices are irreducible.
Let $f$ be the function which maps the group elements to it's corresponding matrix as defined in $\rho$. We could have more interesting functions but I am trying to keep it simple.
So, the Fourier transform of $f$, $\hat{f}$ is given below.
$$ \hat{f} (\rho) = \sqrt{\frac{d_\rho}{|S_3|}} \sum_{s\in S_3} f(s) \rho(s^{-1}) \\ = \sqrt{\frac{3}{6}} \sum_{s\in S_3} f(s) \rho(s^{-1}) \\ = \sqrt{\frac{1}{2}} \left( f(e) \rho(e^{-1}) + f((1,2)) \rho((1,2)^{-1}) + f((2,3)) \rho((2,3)^{-1}) + f((1,3)) \rho((1,3)^{-1}) + f((1,2,3)) \rho((1,2,3)^{-1}) + f((1,3,2)) \rho((1,3,2)^{-1})\right) \\ = \sqrt{\frac{1}{2}} \left( f(e) \rho(e) + f((1,2)) \rho((1,2)) + f((2,3)) \rho((2,3)) + f((1,3)) \rho((1,3)) + f((1,2,3)) \rho((1,3,2)) + f((1,3,2)) \rho((1,2,3))\right) \\ = \sqrt{\frac{1}{2}} \left( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ 1 & 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}\right) \\ =\left( \begin{array}{ccc} \sqrt{2} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \sqrt{2} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \sqrt{2} \\ \end{array} \right) $$
My question:
Did I do it right?