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I would like to obtain explicit matrices for the representations of the Symplectic group $Sp_2(Z)$. For a pair of weights $(a, b)$ I know that the highest weight representations are contained in the tensor product $\otimes^a(M_0) \otimes^b(M_1)$ with two specific representations $M_0$ and $M_1$.

The question is how can I extract the irreducible representation from the tensor product representation? The classical references are mostly glossing over that "detail". Even better is there any software implementation?

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