# Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

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?

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.