A unitary representation of a group $G$ on a Hilbert space $\mathcal{H}$ is a homomorphism $\rho: G \to \mathrm{U}(\mathcal{H})$ (the group of unitary operators) with $\rho(g_1g_2) = \rho(g_1)\rho(g_2)$ and $\rho(e) = I$. A subspace $V \subset \mathcal{H}$ is invariant if $\rho(g)V \subset V$ for all $g$. The representation is irreducible if the only invariant subspaces are $\{0\}$ and $\mathcal{H}$. By Schur's lemma: (1) Any intertwining operator $T: V_1 \to V_2$ between irreducible representations is either zero or an isomorphism; (2) $T: V \to V$ intertwining an irreducible representation with itself is a scalar multiple of the identity.

The tensor product of representations $V_j \otimes V_k$ decomposes into irreducible components: $V_j \otimes V_k \cong \bigoplus_{\ell=|j-k|}^{j+k} V_\ell$. The basis change is given by Clebsch-Gordan coefficients: $|j_1,m_1\rangle\otimes|j_2,m_2\rangle = \sum_J\sum_M \langle j_1 m_1; j_2 m_2 | J M\rangle |J,M\rangle$ where $M = m_1 + m_2$ and $|j_1-j_2| \leq J \leq j_1+j_2$. In physics, this describes the addition of angular momenta: coupling orbital $\ell$ and spin $s$ gives total angular momentum $j$ ranging from $|\ell-s|$ to $\ell+s$.

The character of a representation is $\chi_j(g) = \mathrm{tr}(\rho_j(g))$. For $\mathrm{SU}(2)$: $\chi_j(\theta) = \frac{\sin(2j+1)\theta/2}{\sin\theta/2}$. Characters determine decompositions via the orthogonality relation $\int\chi_j^*(g)\chi_k(g)\,d\mu(g) = \delta_{jk}$. The Wigner-Eckart theorem states that matrix elements of a tensor operator $T_q^{(k)}$ between states $|j,m\rangle$ and $|j',m'\rangle$ factor as $\langle j'm'|T_q^{(k)}|jm\rangle = \langle j'm'|kq;jm\rangle\langle j'||T^{(k)}||j\rangle$ (a Clebsch-Gordan coefficient times a reduced matrix element independent of $m,m',q$).