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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!

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