Let $V$ be a vector space of dimension $n$ over $\mathbb{C}$.
Q.1 When thinking of $V\otimes V\otimes V $, if $V$ has basis $\{x_1,x_2,\cdots,x_n\}$ then can we think $V\otimes V\otimes V$ as $\mathbb{C}$-vector space of polynomials $\sum_{i,j} a_{ij}{x_ix_jx_k}$ where $x_i$ and $x_j$ are non-commuting for $i\neq j$.
If this is true, then consider the natural action of $S_3$ on $V\otimes V\otimes V$. This action makes $V\otimes V\otimes V$ an $\mathbb{C}[S_n]$-module. Consider the element of the group algebra $$\alpha=(1)+(12)+(13)+(23)+(123)+(132).$$ The action of $\alpha$ on $V\otimes V\otimes V$ leaves the subspace $S{\rm ym}^3(V)$ invariant. Now I want to prove that converse is also true:
if an element $\beta\in \mathbb{C}[S_n]$ leaves ${\rm Sym}^3(V)$ invariant, then $\beta$ should some scalar multiple of $\alpha$. How should I proceed for it?
I was thinking elements of $V\otimes V\otimes V$ as in Q.1, but I couldn't proceed for the question on $\beta$
${\rm Sym}^n(V)$ is already a subrepresentation of $V^{\otimes n}$; it is a subspace stabilized by every element of the group algebra. As for elements $\beta$ which fix each element $x$ of ${\rm Sym}^n(V)$, observe
$$ \beta x=\left(\sum_g c(g)g\right)x=\sum_g c(g)gx=\sum_g c(g)x=\left(\sum_g c(g)\right)x. $$
That is, $\beta$ fixes nonzero symmetric tensors $x$ if and only if the sum of its coefficients is $1$. In that case, the sum of the coefficients of $\beta-\alpha$ is $0$, and so $\beta$ fixes them if and only if it is in the coset $\alpha+A$, where $A$ is the augmentation ideal of $\Bbb C[G]$ and $\alpha=|G|^{-1}\sum_g g$ is the normalized sum of the group elements.
Note this argument works on any $\Bbb C[G]$-module, where $V^{\otimes n}$ is replaced with $V$, $S_n$ is replaced with $G$, and ${\rm Sym}^n(V)$ is replaced by the invariant subspace $V^G$ (assuming it's nontrivial).