# Representations of symetric group S3

(sorry in advance for my english. Im not sure my terminology is correct)
I'm trying to solve a problem: Let $e_1$ , $e_2$ , $e_3$ be a base of $C^3$ (3-D vectors with complex elements). Let $A(g)e_i=e_{g(i)}$ be a representation of the symmetric group $S_3$ ($g\in S_3$). Also let $V=\{z_1,z_2,z_3\in C^3|z_1+z_2+z_3=0\}$ a subspace of $C^2$. The question here is to find the restriction ($P(g)$) of $A(g)$ at the subspace $V$, but i don't understand what this means.

## 2 Answers

The problem here is to pick a basis for $V$ and write the matrices for the action of $S_3$ on $V$. The standard basis to choose is $\{\alpha_1,\alpha_2\}$ where $\alpha_1=e_1-e_2$ and $\alpha_2=e_2-e_3$. If we let $s_1=(12)$ and $s_2=(23)$, then $S_3$ is generated by $s_1$ and $s_2$ as a group (e.g. $(123)=s_1s_2$, etc.). Since representations are group homomorphisms, we can determine everything by computing the matrices for the action of $s_1$ and $s_2$. We have $$A(s_1)\alpha_1=s_1(e_1=e_2)=(e_2-e_1)=-(e_1-e_2)=-\alpha_1$$ and $$A(s_1)\alpha_2=s_1(e_2-e_3)=(e_1-e_3)=(e_1-e_2)+(e_2-e_3)=\alpha_1+\alpha_2.$$ Thus, $$P(s_1)=\begin{bmatrix}-1&1\\0&1\end{bmatrix}.$$ Similarly, you can compute that $$P(s_2)=\begin{bmatrix}1&0\\1&-1\end{bmatrix}.$$ You can now get the rest of the matrices by multiplying these together as appropriate. For example \begin{align} P((123))&=P(s_1s_2)\\&=P(s_1)P(s_2)\\ &=\begin{bmatrix}-1&1\\0&1\end{bmatrix}\begin{bmatrix}1&0\\1&-1\end{bmatrix}\\ &=\begin{bmatrix}0&-1\\1&-1\end{bmatrix} \end{align}

Since $V$ is an invariant subspace of $\mathbb C^3$, $\rho(g)(v) \in V$ for every $v \in V$. Thus you have to compute the $2 \times 2$ matrix associated to every element $g \in S_3$.