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Example
The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the tensor-product representation of two copies of the defining representation, we let it act on the tensor-product space as shown in the lower branch.
The resulting 4-dimensional representation is reducible, breaking up into a 1- and 3-dimensional irreducible representation.
This representation is useful for describing a system of two spin-1/2 particles, in particular, to analyze its combined spin.
For a more detailed explanation of this diagram see Fun with Symmetry.