We describe a general theoretical framework based on the Bergman spectral representation to study how a nanostructure interacts with an external electromagnetic field. The selection rules for localized surface plasmon resonances (LSPRs) are obtained by implementing the group theory upon the electric vector field. The influence of symmetry breaking on the splitting of degenerated modes and the switching of dark modes by specific illuminations are discussed. These results emphasize the fact that the selection rules for a vector field are different from the case of a scalar field and essentially induced by the geometry of the structure. Finally, this work not only points out that measurements of LSPRs may result in very different results with different external fields, but also provides a strategy to selectively excite specific LSPRs of plasmonic structures.