Let $e_1$ and $\psi$ be two polynomials, and consider the subspace $V$ of $\mathbb{C}[t,u,v]$ spanned by $e_1(x)\psi(y)$ where $x,y\in \{t,u,v\}$. Then $V$ carries a representation of the symmetric group $S_3$ by permutation on the three variables $\{t,u,v\}$, and we want to reduce the representation $V$. The group $S_3$ has 3 irreducible representations (trivial, alternating and standard) with respective dimensions $1$, $1$ and $2$.
I did the following:
Reduction is made according to the group chain: $S_3 \supseteq S_2$ and intrinsic group of $S_2$. For $S_3 \supseteq S_2$ I defined class operator $D(S_3)=D(tu)+D(tv)+D(uv)$ and $D(S_2)=D(tu)$. And for intrinsic group of $S_2$ $D(s_2)=D(12)$. $D(*)$ represent matrix representation and, for example, $(tu)$ represent permutation where $v$ is unchainged and $t$ is permutated to $u$ and vise versa. Set of operators $(D(S_3),D(S_2),D(s_2)$ is complite set of commutating operators.
From this we get eigen equations: $$D(S_3) \psi_{(\lambda, m, k)}=\lambda \psi_{(\lambda, m, k)}$$ $$D(S_2) \psi_{(\lambda, m, k)}=m \psi_{(\lambda, m, k)}$$ $$D(s_2) \psi_{(\lambda, m, k)}=k \psi_{(\lambda, m, k)}$$ Eigenvalues: $$\lambda=\{-3,0,3\}$$ $$m=\{1,-1\}$$ $$k=\{1,-1\}$$ When eigenvector equations are solved, I get following 9 vectors: $$\psi_{(3,1,1)}^{a}=e_1(t) \psi_1 + e_1(u) \psi_2+e_1(v)\psi_3$$ $$\psi_{(3,1,1)}^{b}=e_1(v) \psi_1 + e_1(t) \psi_2+e_1(u)\psi_3+e_1(u) \psi_1 + e_1(v) \psi_2+e_1(t)\psi_3$$ $$\psi_{(3,-1,-1)}=e_1(v) \psi_1 + e_1(t) \psi_2+e_1(u)\psi_3-e_1(u) \psi_1 - e_1(v) \psi_2-e_1(t)\psi_3$$ $$\psi_{(0,-1,k)}=e_1(t) \psi_1 - e_1(u) \psi_2$$ $$\psi_{(0,1,k)}=e_1(t) \psi_1 + e_1(u) \psi_2-2e_1(v)\psi_3$$ $$\psi_{(0,-1,1)}=e_1(v) \psi_1 -e_1(u)\psi_3- e_1(v) \psi_2+e_1(t)\psi_3$$ $$\psi_{(0,1,1)}=e_1(v) \psi_1 -2 e_1(t) \psi_2+e_1(u)\psi_3-2e_1(u) \psi_1 + e_1(v) \psi_2+e_1(t)\psi_3$$ $$\psi_{(0,1,-1)}=-e_1(v) \psi_1 +e_1(u)\psi_3 - e_1(v) \psi_2+e_1(t)\psi_3$$ $$\psi_{(0,-1,-1)} =-e_1(v) \psi_1 +2 e_1(t) \psi_2-e_1(u)\psi_3-2e_1(u) \psi_1 + e_1(v) \psi_2+e_1(t)\psi_3$$
$\psi_{(3,1,1)}^{a}$ and $\psi_{(3,1,1)}^{b}$ form trivial representation of $S_3$ while $\psi_{(3,-1,-1)}$ form alterneiting representation. Remaining 6 vectors span 3 standard two dimensional representations.
After that, I defined similarity transformation matrix $T$ and got $(tuv)$ permutation matrix in new basis: \begin{equation} (tuv)_{2}= T_1(tuv)_1 {T_1}^{-1}=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -3/2 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1/2 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 3/2 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & -1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & -1/2\\ \end{bmatrix} \end{equation} Finding $(tu)$ matrix is analogous and since this 2 permutations span whole group $S_3$ other matrices can be obitained trivially with multiplications of this 2 matrices.
Questions are following: I belive this is correct solution but I have trouble finding good literature. Where can I read more about this? Most books I found about representation theory of groups are too difficult to understend for me. Are all 3 standard representations which were obitened equivalent? Can they be named differently?
On https://groupprops.subwiki.org/wiki/Standard_representation_of_symmetric_group:S3 I found following matrix element for standard representation: \begin{equation} \begin{bmatrix} -1/2 & -\sqrt{3}/2\\ \sqrt{3}/2 & -1/2\\ \end{bmatrix} \end{equation} for $(123)$ permutation and my matrix element looks like this: \begin{equation} \begin{bmatrix} -1/2 & -1/2\\ 3/2 & -1/2\\ \end{bmatrix} \end{equation} I did not write all other matrices here but when I check traces of matrices (characters) they are all as they should be. What rises a confusion is this $\sqrt{3}$. Did I make mistake in my calculation? I have a lot of question about this so best solution for me is to get a good book which can help me. I am hoping someone will recognize at least something and give me good direction on where to search for solution. I never had any classes about reduction of groups at my university so a lot of this is completly new for me. Thank you for your time.