# characters of symmetric and exterior algebras [duplicate]

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.

## marked as duplicate by darij grinberg, José Carlos Santos, YuiTo Cheng, Leucippus, CesareoMay 4 at 8:36

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