The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.

So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).

I'd like to find the decomposition into irreps of it. As I'm a physicist, I'm first interested in the simple cases $p=2$ (and if this is still difficult, $n=3,4$), even if something general would be nice!

I also need something constructive: I guess the proof will be constructive, but I'd prefer a reference where the basis vectors are easily tractable.

Thanks!