On tensor induction of group representations

Abstract

AbstractLetGbe a (not necessarily finite) group andρa finite dimensional faithful irreducible representation ofGover an arbitrary field; writeρ¯forρviewed as a projective representation. Suppose thatρis not induced (from any proper subgroup) and thatρ¯is not a tensor product (of projective representations of dimension greater than 1). LetKbe a noncentral subgroup which centralizes all its conjugates inGexcept perhaps itself, writeHfor the normalizer ofKinG, and suppose that some irreducible constituent, σ say, of the restrictionp↓Kis absolutely irreducible. It is proved that then (ρis absolutely irreducible and) ρ¯ is tensor induced from a projective representation ofH, namely from a tensor factor π ofρ¯↓Hsuch that π↓K= σ¯ and ker π is the centralizer ofKinG.