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Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix.

The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots \end{pmatrix} \quad \begin{pmatrix}0 \\1 \\ \vdots \end{pmatrix} \quad \ldots$$

How can I use this to compute the basis vectors of the $54$ dimensional representation, written as $10 \times 10 $ matrix?

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Simply use the Kronecker product $e_i \otimes e_j$ to compute the basis vectors, which are then matrices for the product representation. Then symmetrize and antisymmetrize to get basis vectors for the corresponding irreducible representations.

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