Representation

The representation of a Group $G$ on a Complex Vector Space $V$ is a group action of $G$ on $V$ by linear transformations. Two finite dimensional representations $\pi$ on $V$ and $\pi'$ on $V'$ are equivalent if there is an invertible linear map $E:V\mapsto V'$ such that $\pi'(g)E=E\pi(g)$ for all $g\in G$. $\pi$ 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

References

Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.