I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the standard representation in a basis where $\underline{e}_1 = \sqrt{\frac{2}{3}} (1, -\frac{1}{2}, -\frac{1}{2})$, $\underline{e}_2 = \sqrt{\frac{1}{2}}(0,1,-1)$ and $\underline{e}_3 = \frac{1}{\sqrt{3}}(1,1,1)$
I can obtain the corresponding transformation matrix and then compute $\Gamma' = T^{\dagger}\Gamma T = T^T \Gamma T$, but when I have this $\Gamma'$ I don't know how to then extract the standard representation as per the question requirements.
Many thanks.
I won't calculate the exact result but let me give you some ingredients:
You should know how to transform a matrix to another via a change of basis. This way we can work with a preferred one. Mine is the standard basis. Since you used $e_i$ already lets call them $v_1,v_2,v_3$.
You hopefully know how the permutation matrices that you called $\Gamma((1,2)),\dots$ look like. (e.g. $\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}$)
Now the part that you probably had trouble with. The standard representation lives on the vector space usually defined by $\mathbb C^3/ \langle(1,1,1)\rangle$ (assuming you work over $\mathbb C$ although every field would work here). This vector space is two-dimensional and one can see that the images of $v_1$ and $v_2$ are again form a basis. Let us call them $\bar v_1, \bar v_2$.
Now we only need to know what happens on these two vectors. Note that the image of $v_1+v_2+v_3$ is zero, so whenever we get $\bar v_3$ we replace it by $-\bar v_1-\bar v_2$.
Example: $\Gamma((1,2,3))$
$v_1$ is sent to $v_2$ and $v_2$ is sent to $v_3$, hence $\bar v_1$ gets sent to $\bar v_2$ and $\bar v_2$ gets sent to $\bar v_3 = -\bar v_1-\bar v_2$. Thus the matrix is $\begin{pmatrix}0&1\\-1&-1\end{pmatrix}$.
Now you only have to work out what the images $\bar e_1, \bar e_2$ of $e_1,e_2$ are in terms of $\bar v_1, \bar v_2$ and do the base change.