I'm currently self studying Fulton Harris, unfortunately I don't really know where to check my answers, and I don't know if this logic is correct:
Notation: Let $U$, $U'$, $V$ be the trivial, alternating, standard representations of $S_3$ respectively, and $\omega = e^{2\pi i / 3}$. We already have a basis for $V$ that being $\alpha = (\omega, 1, \omega^2)$ and $\beta = (1, \omega, \omega ^2)$. We also have two transpositions $\sigma = (1 2)$ and $\tau = (1 2 3)$ which generate $S_3$, and these properties were already established:
\begin{align*} \tau \alpha =& \omega \alpha\\\\ \tau \beta =& \omega^2 \beta\\\\ \sigma \alpha =& \beta\\\\ \sigma \beta =& \alpha \end{align*}
And right before this exercise it's also established that: $$V \otimes V \cong U \oplus U' \oplus V$$ By using the basis $\\{\alpha \otimes \alpha,\alpha \otimes \beta,\beta \otimes \alpha,\beta \otimes \beta\\}$ And noticing that $\\{\alpha \otimes \alpha, \beta \otimes \beta\\}$ spans $V$ and $\alpha \otimes \beta + \beta \otimes \alpha$ spans $U$, and $\alpha \otimes \beta - \beta \otimes \alpha$ spans $U'$.
Exercise 1.11: Find a decomposition of the representations $Sym^2 V$ and $Sym^3 V$.
My answer: Since $Sym^2 V \leq V \otimes V$, the previous basis spans $Sym^2 V$, in particular since the action of $S_2$ on $\langle \alpha\otimes \alpha, \beta \otimes \beta\rangle$ is trivial so it's isomorphic to $V$. The span of $\langle \alpha \otimes \beta, \beta \otimes \alpha \rangle$ is one dimensional as $\alpha \otimes \beta = \beta \otimes \alpha$, and is thus equal to $\langle \alpha \otimes \beta + \beta \otimes \alpha\rangle = U$, so $Sym^2 V \cong U \oplus V$.
On $Sym^3 V$ we have a spanning set: $\\{\alpha \otimes \alpha \otimes \alpha,\alpha \otimes \alpha \otimes \beta, \alpha \otimes \beta \otimes \beta, \beta\otimes \beta \otimes \beta \\}$, we notice that by setting $v = \alpha \otimes \alpha \otimes \alpha$, we have $\sigma(v) = \beta \otimes \beta \otimes \beta$. And that both $v$ and $\sigma(v)$ are eigenvectors of $\tau$ with eigenvalue 1, and that $\langle \sigma(v) + v\rangle \cong U$ and $\langle \sigma(v) - v \rangle \cong U'$. By setting $w = \alpha \otimes \alpha \otimes \beta$, then $\sigma(w) = \alpha \otimes \beta \otimes \beta$, and $\tau(w) = \omega w$ and $\tau(\sigma(w)) = \omega^2 \sigma(w)$, and $\langle \sigma(v), v \rangle \cong V$, and therefore $Sym^3 V \cong U \oplus U' \oplus V$.
I'd be very grateful if anyone could check if this is correct, as this is my first exposure to symmetric powers.