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We study multipole decompositions of the electromagnetic currents ofspin-1/2, 1, and 3/2 particles described in terms of Lagrangians designed toreproduce representation specific wave equations which are second order in themomenta and which emerge within the recently elaborated Poincar\'e covariantprojector method. We calculate the electric multipoles of the above spins forthe spinor, the four-vector, and the four-vector--spinor representations,attend to the most general non-Lagrangian spin-3/2 currents which are allowedby Lorentz invariance to be of third order in the momenta and construct thelinear current equivalent of identical multipole moments of one of them. Weconclude that such non-Lagrangian currents are not necessarily more generalthan the two-term currents emerging within the covariant projector method. Wecompare our results with those of the conventional Proca-, and Rarita-Schwingerframeworks. Finally, we test the representation dependence of the multipoles byplacing 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 representationindependence of the charge monopoles and the magnetic dipoles, in contrast tothe higher multipoles, which turn out to be representation dependent. Inparticular, we find the bi-vector $(1,0)\oplus (0,1)$ to be characterized by anelectric quadrupole moment of opposite sign to the one found in $(1/2,1/2)$,and consequently, to the $W$ boson. Our finding points toward the possibilitythat the $\rho$ meson could transform as part of an antisymmetric tensor withan $a_{1}$ meson-like state as its representation companion.

Electromagnetic multipole moments of elementary spin-1/2, 1, and 3/2 particles