Abstract: Character table consisting of irreducible representations (IRs) is the basis of the group theory, and the orthonormality in irreducible representations has been proved by the great orthogonality theorem (GOT). In teaching group theory, we introduce Dirac symbols to represent irreducible representations, in order to provide a concise and easily understanding expression of the GOT. Consequently, the completeness of the character space, expanded by the irreducible representations, is expressed accordingly. On the basis of the completeness of character space, the reducible representations of the normal mode of vibrations of two molecules (H2O and NH3) are expanded by the irreducible expansions, in order to illustrate the applications of Dirac symbols in group theory. Such teaching plan is expected to help students to better understand molecular point group, symmetry operations, irreducible representations, the GOT, completeness of character space, and the decomposition of reducible representation, toward improving the quality of teaching.

Key words: great orthogonality theorem, irreducible representations, Dirac symbols, character table, completeness