characters of symmetric and exterior algebras

Question

Let $\rho,V$ be a representation of $S_3$.

In my lecture notes, It says that we can deduce the characters of $V\otimes V$ and $\operatorname{Sym}^2(V)$ and $\bigwedge^2V$ only from the character table of $S_3$.

I don't see how for the symmetric and exterior algebras, as those are quotients and I am not aware of operations on character of quotient representations.

Thank you for your hints or help.

Edit

Trying to do the exercise pointed out by @darij in Alex's Answer.

Let $G=S_3=\{\sigma, \tau|\sigma^2=\tau^3=\sigma\tau\sigma\tau=e\}$.

Taking the notations of the other question

We have $m=2$ and $g=\tau$ and we try to find $\chi_{Sym^2(V)}$.

Let $1,j,j^2$ be eigenvalues of $\tau$ and $1, -1$ those of $\sigma$. I don't understand the step 2. How do we get $\chi_{Sym^2(V)}$ in terms of the eigenvalues?

Thank you.

Answer 1

I see now what Alex meant in his answer:

Similarly to his formulae for $Sym^3$ and $\wedge^3$, we have:

$\chi_{\text{Sym}^2V}(g) = \frac{\chi(g)^2 + \chi(g^2)}{2}$ and $\chi_{\Lambda^2V}(g) = \frac{\chi(g)^2 - \chi(g^2)}{6}$.