The representation of a Group on a Complex Vector Space is a group action of on by linear transformations. Two finite dimensional representations on and on are equivalent if there is an invertible linear map such that for all . is said to be irreducible if it has no proper Nonzero invariant Subspaces.
See also Character (Multiplicative), Peter-Weyl Theorem, Primary Representation, Schur's Lemma
Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.