Irreducible representation of $S_3$ by the space of homogeneous polynomials of order 3. I am studying representation theory and got stuck on the following problem.
Let $V$ be the vector space of degree 3, real homogeneous polynomials of three variables, then $S_3$ acts on $V$ by permuting the variables. Then, describe the decomposition of $V$ into irreducible representations of $S_3$. 
Where do I even get started with this? 
 A: Start by checking how the identity, the 3-cycle $(1 2 3)$ and the 2-cycle
$(1 2)$ operate on the basis of $V$ given by
$$
x_{1}^{3}, x_{2}^{3}, x_{3}^{3},\\
x_{1}^{2} x_{2},  x_{2}^{2} x_{1},\\  
x_{1}^{2} x_{3},  x_{3}^{2} x_{1},\\  
x_{2}^{2} x_{3},  x_{3}^{2} x_{2},\\  
x_{1} x_{2} x_{3}.
$$
For instance, $(1 23)$ cyclically permutes the first three elements.
This will yield matrices for the actions of the identity, the 3-cycle $(1 2 3)$ and the 2-cycle
$(1 2)$, and thus the character $\chi$ of the representation.
Then it is not difficult to use the usual inner product to compute the multiplicities of the irreducible characters of $S_{3}$ in $\chi$.

So you get the character $\chi$
$$
\begin{matrix}
1 & (123) & (12)\\\hline
10 & 1 & 2
\end{matrix}
$$
meaning that it has value $10$ on the identity, value $1$ on the elements of the conjugacy class of $(123)$, and $2$ on the elements of the conjugacy class of $(12)$.
Now the character table of $S_{3}$ is
$$
\begin{matrix}
 & 1 & (123) & (12)\\\hline
\chi_{1} &
1 & 1 & 1\\
\chi_{2} &1 & 1 & -1\\
\chi_{3} &2 & -1 & 0\\
\end{matrix}
$$
Now compute the inner product of $\chi$ with the $\chi_{i}$ to get the multiplicities of each $\chi_{i}$ in $\chi$.
