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Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.

Associated to $\chi$ is a projection from $W$ onto a subspace, which we will denote by $\pi_\chi$ (this is standard, but I can write down the formula for $\pi_\chi$ if needed). We will also denote by $\chi$ the element of the group algebra $\mathbb{C}[S_n]$ corresponding to $\pi_\chi$. Let us now form $$ \rho = \sum_{h \in H} h \chi h^{-1} \in \mathbb{C}[S_n].$$ Note that, for any $h \in H$, we have $h \rho = \rho h$.

I am interested in the subspace of $W$ obtained as the image of $\rho$. How can one decompose it into irreducibles, under the action of $\operatorname{SL}(2, \mathbb{C})$? Has it been studied in the literature?

Edit 1: it turns out that actually, the $\rho$ I am interested in is of the form $$ \rho = \sum_{h \in H} h \chi \in \mathbb{C}[S_n],$$ so there is no conjugation, unlike what I had previously thought. @Will Sawin suggested that I apply the Schur-Weyl duality, which is indeed the right tool to use for this question.

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    $\begingroup$ Schur-Weyl duality lets us write $W$ as a sum of tensor products of irreducible representations of $SL_2$ and irreducible representations of $S_n$ in an explicit way. So your question about decomposition to irreducible representations of $SL_2$ reduces to calculating the rank of $\rho$ when restricted to these irreducible representation of $S_n$. This is clearly calculable explicitly in terms of $G$ and $H$ so the only thing to ask for is some kind of closed-form formula. I don't expect such a thing exists, although I can't disprove it. $\endgroup$
    – Will Sawin
    Commented Apr 3 at 20:28
  • $\begingroup$ Yes, I don't understand the Schur-Weyl duality sufficiently well right now. I will spend some time on that then! Thank you. $\endgroup$
    – Malkoun
    Commented Apr 3 at 22:11
  • $\begingroup$ I took an example to understand the Schur-Weyl duality and it finally made sense to me. Thank you! It is interesting that one can look at the decomposition you have mentioned as a representation of $SL(2, \mathbb{C})$ alone, or as a representation of $S_n$ alone. I know, this is why it is called "duality", but I had to work out an example to understand it. Nice. As far as this post goes, could you perhaps write your comment as an answer? I am interested in a specific example of $G$ and $H$, but I guess I can work it out now. Thank you! $\endgroup$
    – Malkoun
    Commented Apr 4 at 1:18

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