We introduce new methods from representation theory of algebraic groups into the study of representation zeta functions associated with compact $\mathfrak{p}$-adic analytic groups and arithmetic groups. We apply these new methods to compute the representation zeta functions of principal congruence subgroups of the groups $\mathrm{SL}_4(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic 0.