Some Math

Aug 11, 2025

Definition

Given a group $G$ with symmetry elements $g$ and symmetry operators $\hat{P}_g$ , we denoted the irreducible representations by $\Gamma_n$ , where the $n$ labels each different irreducible representation.

We then define a set of basis vectors for each representation denoted by $\ket{\Gamma_n j}$ , where the $j$ index labels the so-called component or partner of a representation. The index $j$ runs from $1$ to $\ell_n$ , the dimension of the representation.

The partners collectively generate the matrix representation of $\Gamma_n$ , denoted by $D^{(\Gamma_n)}(g)$ , via

$\hat{P}_g \ket{\Gamma_n \alpha} = \sum_{j} D^{(\Gamma_n)}(g)_{j\alpha} \ket{\Gamma_n j}$

Orthogonality Relation

The basis vectors satisfy the orthogonality relation: ¹

$\braket{\Gamma_n j|\Gamma_{n'} j'} = \delta_{jj'} \delta_{nn'}$

Basis Functions

The basis vectors in the most general sense are abstract vectors, but they can also be basis functions, which we define in this context as basis vectors expressed directly in coordinate space. Wavefunctions in quantum mechanics are such an example of basis functions of symmetry operators. [1,2].

In this case, we have: [3]

$\int \psi_{n,j}^*(\pmb{r}) \psi_{n'j'}(\pmb{r}) \text{d}^3r = \delta_{nn'} \delta_{jj'}$

Here, $n$ labels the energy eigenvalue and $j$ is the degeneracy index within that degenerate subspace.

Generating the matrices for an irrep

Starting from

$\hat{P}_g \ket{\Gamma_n \alpha} = \sum_{j} D^{(\Gamma_n)}(g) \ket{\Gamma_n j}$

We get

$\begin{aligned} \braket{\Gamma_{n'} j'|\hat{P}_g |\Gamma_n \alpha} &= \sum_{j} D^{(\Gamma_n)}(g)_{j\alpha} \braket{\Gamma_{n'} j'|\Gamma_n j}, \\ &= \sum_{j} D^{(\Gamma_n)}(g)_{j\alpha} \delta_{j j'} \delta_{n n'} \end{aligned}$

So we end up with:

$D^{(\Gamma_n)}(g)_{j\alpha} = \braket{\Gamma_{n} j|\hat{P}_g |\Gamma_n \alpha}$

i.e., the matrices for an irrep are just the matrix elements of the symmetry operator $\hat{P}_g$ between all possible partners of an irreducible representation. In practice, this is the easiest way to obtain the matrix representations for the symmetry elements.

Corresponding to a set of basis functions, the matrix representation generated by them is unique. However, basis functions for a representation are not unique. The character is naturally independent of the choice of bais functions.