# Explicit symmetry adapted basis for the symetric square of the standard representation

I already posted a related question here, which is more detailed: https://math.stackexchange.com/posts/2786382/edit

The permutation group $$S_n$$ has standard representation $$S^{(n-1,1)}$$ (irreducible).

Then it naturally acts over $$S^{(n-1,1)}\otimes S^{(n-1,1)}$$.

Under this action, this space decomposes as: $$S^{(n-1,1)}\otimes S^{(n-1,1)}=\Lambda S^{(n-1,1)}\oplus S^{(n)}\oplus S^{(n-1,1)}\oplus S^{(n-2,1^2)},$$

where $$\Lambda S^{(n-1,1)}$$ is the symmetric difference of $$S^{(n-1,1)}$$, and $$S^{(n)}$$ the trivial irreps.

I'm looking for a symmetry adapted basis for this decomposition. I already have it for $$\Lambda S^{(n-1,1)}\oplus S^{(n)}\oplus S^{(n-1,1)}$$: $$\Lambda S^{(n-1,1)}$$ is easy by definition of symmetric difference and $$S^{(n)}\oplus S^{(n-1,1)}$$ can be seen as the natural representation embeded in $$S^{(n-1,1)}\otimes S^{(n-1,1)}$$. I did not managed $$S^{(n-2,1^2)}$$ easily, any nice expression (I could take the orthogonal but I'l like something elegent if possible).

I have two extra question:

• what is the notation for $$\Lambda S^{(n-1,1)}$$?

• is there a package on matlab, pyton, ... to give this sort of basis?

Thanks a lot!