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The Schur-Weyl duality states that $\bigotimes_{m=1}^n \mathbb{C}^k$ can be decomposed as a direct sum over the tensor product of irreductible representations of $SU(k)$ and of the symmetric group $\mathfrak{S}_n$.

If one considers the semi-normal representations of $SU(k)$ (resp. $\mathfrak{S}_n$), the irreducible representations of this groups are equipped with a basis indexed by semi-standard (resp. standard) Young tableaux whose shape $\lambda$ indexes the representation.

Putting everything together, we thus have a decomposition: $$ \bigotimes_{m=1}^n \mathbb{C}^k = \bigoplus_{(S,T) \text{ with shape $\lambda$ with $n$ boxes and at most $k$ lines}} (v_{S}\otimes v_T) $$ and it should give a change of basis from the canonical basis of $\bigotimes_{m=1}^n \mathbb{C}^k$ to the basis $v_S\otimes v_T$ (on which $su(k)$ and $\mathfrak{S}_n$ acts in an easy way).

My question is (at least for small values of $k$ and $n$) : where can I find the matrix of the change of basis ? How can I compute it easily ?

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  • $\begingroup$ In quantum computing this change of basis is known as Schur transform. In principle, it can be obtained as a sequence of so-called Clebsch–Gordan transforms but it is quite tedious. Some relevant references: Jakubczyk et al., Bacon et al., and Chapter 5 of Harrow's thesis. $\endgroup$ – Māris Ozols Aug 14 '17 at 12:34

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