# Definition Given a [[group]] $G$ with symmetry [[Group Element|elements]] $g$ and symmetry operators $\hat{P}_g$, we denoted the [[Irreducible Representation|irreducible representations]] by $\Gamma_n$, where the $n$ labels each different irreducible representation. We then define a set of *[[basis]] [[vector|vectors]]* for each [[Group Representation|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: $ \braket{\Gamma_n j|\Gamma_{n'} j'} = \delta_{jj'} \delta_{nn'} $ # Basis Functions The basis vectors in the most general sense are abstract [[vector|vectors]], but they can also be *basis functions*, which we define in this context as basis vectors expressed directly in [[real coordinate space|coordinate space]]. [[Wavefunction|Wavefunctions]] in quantum mechanics are such an example of basis functions of symmetry operators. In this case, we have: $ \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 [[vector subspace|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{align} \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{align} $ 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.