Group Representation
Definition. A homomorphism $\rho: G \to GL(V)$ — the group acting linearly on a vector space $V$. Irreducible if $V$ has no invariant proper subspace. Unitary if $\rho(g)$ preserves an inner product.
Working examples. U(1): all irreducibles are 1-dimensional, $\rho_n(e^{i\theta}) = e^{in\theta}$, $n \in \mathbb{Z}$. SU(2): one irreducible of each dimension $2j+1$, $j = 0, \tfrac12, 1, \dots$ — spin.
Intuition (Woit's central theme). Quantum systems carry representations of their symmetry groups; particle labels (charge, spin, isospin, color) are representation labels. Woit's book is one long elaboration of this idea.
Role in bundle theory. A representation is the input to the Associated Bundle construction: principal bundle $P$ + rep $\rho$ ⇒ vector bundle $P \times_\rho V$ whose sections are matter fields "of type $\rho$." Charge $n$ = the field lives in $\rho_n$.
Related: U(1) · SU(2) and SO(3) · Adjoint Representation · Associated Bundle · Module 10 - Representations and Matter Fields
Sources: Woit (throughout; esp. $U(1)$ and $SU(2)$ chapters).