I am reading Fulton and Harris and on page 150, there is the following passage (in the second paragraph) that I don't understand:
"Similarly, a basis for the symmetric square $W = \textrm{Sym}^2V = \textrm{Sym}^2 \Bbb{C}^2$ is given by $\{x^2,xy,y^2\}$ and we have $\begin{eqnarray*} H(x \cdot x) &=& x \cdot H(x) + H(x) \cdot x = 2 x \cdot x\\ H(x\cdot y) &=& x\cdot H(y) + H(x) \cdot y = 0\\ H(y \cdot y) &=& y \cdot H(y) + H(y) \cdot y = 2y \cdot y\end{eqnarray*}$ so the representation $W = \Bbb{C}\cdot x^2 \oplus \Bbb{C}\cdot xy \oplus \Bbb{C}\cdot y^2 = W_{-2} \oplus W_0 \otimes W_2$ is the representation $V^{(2)}$ above."
I should say $x,y$ are the standard basis vectors of $\Bbb{C}^2$ and $H$ is the matrix $diag(1,-1)$.
My question: What are $W_0$, $W_{-2}$ and $W_2$ in the last line of that paragraph? Also, is it a typo where they write the tensor product of $W_0$ and $W_2$ instead of direct sum? Furthermore, what is this $V^{(2)}$ mentioned? I have looked through the text and there is mentioned things like $V_\alpha$, but none with superscripts.