Some Math

Definition Given a group G with symmetry elements g and symmetry operators P̂_(g), we denoted the irreducible representations by Γ_(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...

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Aug 11, 20252 min read

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. [@item1; @item2].

In this case, we have: [@freitasReliabilityEntropyProduction2022]

\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{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.

Example Plot

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By the way this is a horrible idea ↩

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