An intuitive approach to group representation theory

The purpose of this article is to outline a nonrigorous and highly intuitive introductory approach to the sort of group representation theory used in molecular and solid state physics. We examine the real three‐space spanned by orthonormal polar basis vectors {‖i〉, i=1,2,3}; first it is shown that the familiar block diagonal ’’rotation matrix’’ is simply the {‖i〉} representation of the geometric transformation operator ?_φ which causes a counterclockwise rotation of any three‐space polar vector by the angle φ about the ‖3〉 axis. By applying this argument to the group of symmetry operators of the ammonia molecule, it becomes self‐evident that the two irreducible representations obtainable from the {‖i〉} representation of C_3v correspond to subspaces of real‐space invariant relative to the six operators of that group. And the basis vectors which span these invariant subspaces (and which produce the corresponding irreducible representations) have clearly defined symmetry (A₁ and E) properties under the operations of C_3v. This idea is further illustrated in consideration of the representation produced by the in‐plane normal modes of vibration of the three hydrogen atoms in ammonia. Finally, it is shown that the approach can readily be extended into function space.