Consider the direct product space of two angular momentum eigenfunctions:
$$|j_1, j_2; m_1, m_2⟩ = |j_1, m_1⟩|j_2, m_2⟩$$
for the simple case when
$$j_1 = j_2 = 1/2.$$
How can i construct the direct product space explicitly and make the connection with group representation theory?
The direct product space is spanned by the 4 states $\vert{s_1m_1}\rangle\vert{s_2,m_2}\rangle$. This Hilbert space carries a representations of $\frac{1}{2}\otimes\frac{1}{2}$, and group theory tells you this representation is reducible as $1\oplus 0$.
Moreover, group theory explicitly helps in constructing states which reduce this representation since the search for irreducible subspaces amounts to a search for highest weight states, i.e. states killed by the action of the (here unique) raising operator so that $J_+\vert{j,j}\rangle=0$.
Once you have the highest weight you can generate the rest of the irreducible piece by the action of $J_-$.
