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This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$ for both cases when 1) p is odd 2) p-is even, here $\mathbf(V)$ is of sufficiently high dimension. This can be achieved by applying the duality and using the Cayley-Hamilton theorem (as you can see from the nice answers I got).

Now, I would like to find explicit construction of these irreducible spaces. That is given an irreducible representation $\mathbb{S}_\lambda \mathbf(V) \subset Sym^m(\bigwedge^p \mathbf(V))$ characterised by a Young-diagram $\lambda$ with 2 or less columns with multiplicity $m_\lambda$, I would like to find what weight vectors generate them for each of the copies of equivalent irreducibles.

I guess this will be given by the action of the young symmetriser for a subset of(equal to multiplicty) tableaux generated by the Littlewood richardson for the tensor product $(\bigwedge^p \mathbf(V))^{\otimes m}$ since

$Sym^m(\bigwedge^p \mathbf(V)) \subset (\bigwedge^p \mathbf(V))^{\otimes m} $

I realise that this might be a difficult problem in general, but in this case this it is very specialised, and only 2 or less number of columns are considered.

I will appreciate if anyone can provide an algorithm or positive rule or point me to a reference.

(p.s A brute force approach is to write down all tableaux for given young diagram, given by littlewood and symmetrise the collection of factors and see which one survives.)

I am a physicist and I apologise if this is a silly question.

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