I am reading Fulton and Harris's representation theory and not sure about something.
$\frak{sl}_2(\mathbb{C})=$$span\{H,X,Y\}$,where $H=\begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix}$, $X=\begin{bmatrix} 0 &1 \\ 0 &0 \end{bmatrix}$ and $Y=\begin{bmatrix} 0 &0 \\ 1 &0 \end{bmatrix}$.
Let $V=V_{-1}\oplus V_1$, where $V_\alpha$ is egienspace with root $\alpha$.Here dimension of $V$ is 2 and $V$ is irreducible representation.
Let $W=Sym^2(V)$. In the book, the author claimed that $H(x\cdot x)=x\cdot H(x)+H(x)\cdot x=2x\cdot x$.
However, I am not sure. Given two arbitrary vector space $A,B$ and linear map $f:A\rightarrow B$. We can define $F:Sym^2(A) \rightarrow Sym^2(B)$ by $F(a_1a_2)=f(a_1)f(a_2)$.
If we pick $A=B=V$, then $H(x\cdot x)=H(x)H(x)=x\cdot x$. I don't understand how he get $H(x\cdot x)=2x^2$.
Thank you!