TITLE: Decomposition of representation of symmetric group
QUESTION [5 upvotes]: Let $V$, $\dim V=n-1$  be the standard representation of the symmetric group $S_n$  and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its  natural representation.   Then ( see. Fulton, Harris, 4.19) we have 
$$
{\rm Sym}^2 V=U \oplus V \oplus V_{(n-2,2)},
$$
where $U$ is the trivial representation and $V_{(n-2,2)}$  is  a representation that correspond  to a partition $(n-2,2).$
Question 1.  What is the decomposition $
{\rm Sym}^2 V'?
$
Question 2.  Can we indicate in explicit way a basis for each  irreducible component of this decomposition?
My try is as follows. Since $V'=U \oplus V$  then 
$$
{\rm Sym}^2 V'={\rm Sym}^2U \oplus {\rm Sym}^2 V \oplus U \cdot V.
$$
I think that ${\rm Sym}^2U \cong U $  and $U \cdot V \cong V.$
Then 
$$
{\rm Sym}^2 V'={\rm Sym}^2U \oplus {\rm Sym}^2 V \oplus U \cdot V=2 U \oplus 2 V \oplus V_{(n-2,2)}.
$$
Am I right? 
If yes,  then what be answer  for the second question? I know  that $U=\langle x_1+x_2+\cdots+x_n \rangle$ and $V= \langle x_1-x_2,x_1-x_3,\ldots,x_1-x_n \rangle$,  but I don't know  what is  the basis for its realisation in ${\rm Sym}^2V'$  as polynomials  of degree $2.$
Edit.
$2U=\langle x_1^2+x_2^2+\cdots+x_n^2 \rangle \oplus \langle x_1 x_2+x_1 x_2 +\cdots+ x_{n-1} x_n\rangle $
Butwhat is  the basis of $V_{(n-2,2)}$ and $V$  realised in ${\rm Sym}^2 V'?$

REPLY [2 votes]: You're right, that's the decomposition.
As you mention,
$Sym^2(U\oplus V)=Sym^2U\oplus (U\cdot V) \oplus Sym^2V$. From this you can immediately see that $Sym^2(U)$ corresponds to $\langle (x_1+...+x_n)^2\rangle$, and $U\cdot V$ to
$(x_1+...+x_n)\langle x_1-x_2,...,x_{n-1}-x_n \rangle$. The copies of the trivial and standard representations inside $Sym^2(V)$ correspond to $U\cong\langle \sum_{i,j}(x_i-x_j)^2\rangle$ and $V\cong \langle (nx_i-\sum_{j=1}^nx_j)^2-(nx_{i+1}-\sum_{j=1}^nx_j)^2 \rangle_{i=1}^{n-1}$. For the copy of $V_{(n-2,2)}$ in theory you can apply the proof of Maschke's theorem to find the complement to $U\oplus V$ in $Sym^2(V)$.