We study multipole decompositions of the electromagnetic currents of spin-$1/2$, 1, and $3/2$ particles described in terms of representation-specific wave equations which are second order in the momenta and which emerge within the recently elaborated Poincaré covariant-projector method, where the respective Lagrangians explicitly depend on the Lorentz group generators of the representations of interest. The currents are then the ordinary linear Noether currents related to phase invariance, and present themselves always as two-terms motion–plus spin-magnetization currents. The spin-magnetization currents appear weighted by the gyromagnetic ratio $g$, a free parameter in the method which we fix either by unitarity of forward Compton scattering amplitudes in the ultraviolet for spin-1 and spin-$3/2$, or in the spin-$1/2$ case, by their asymptotic vanishing, thus ending up in all three cases with the universal $g$ value of $g=2$. Within the method under discussion, we calculate the electric multipoles of the above spins for the spinor, the four-vector, and the four-vector–spinor representations, and find it favorable in some aspects, specifically in comparison with the conventional Proca and Rarita-Schwinger frameworks. We furthermore attend to the most general non-Lagrangian spin-$3/2$ currents, which are allowed by Lorentz invariance to be up to third order in the momenta and construct the linear-current equivalent of identical multipole moments of one of them. We conclude that nonlinear non-Lagrangian spin-$3/2$ currents are not necessarily more general and more advantageous than the linear spin-$3/2$ Lagrangian current emerging within the covariant-projector formalism. Finally, we test the representation dependence of the multipoles by placing spin-1 and spin-$3/2$ in the respective $(1,0)\oplus (0,1)$ and $(3/2,0)\oplus (0,3/2)$ single-spin representations. We observe representation independence of the charge monopoles and the magnetic dipoles, in contrast to the higher multipoles, which turn out to be representation-dependent. In particular, we find the bi-vector $(1,0)\oplus (0,1)$ to be characterized by an electric quadrupole moment of opposite sign to the one found in $(1/2,1/2)$, and consequently to the $W$ boson. This observation allows us to explain the positive electric quadrupole moment of the $\rho$ meson extracted from recent analyses of the $\rho$ meson electric form factor. Our finding points toward the possibility that the $\rho$-meson could transform as part of an antisymmetric tensor with an $a_1$ mesonlike state as its representation companion, a possibility consistent with the empirically established $\rho$ and $a_1$ vector meson dominance of the hadronic vector and axial-vector currents.

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