- Character Theory in Group Representation
1. Definition of Character
The character of a representation $D(g)$ is defined as the trace of the representation matrix:
$$\chi_D(g) = \text{Tr}(D(g))$$
2. Orthogonality Relations
From the orthogonality relation for matrix elements:
$$\frac{n_a}{N}\sum_{g \in G} [D_a(g)]^*_{jk}[D_b(g)]_{lm} = \delta_{ab}\delta_{jl}\delta_{km}$$We can derive the orthogonality relation for characters:
$$\frac{n_a}{N}\sum_{g \in G, j, l} [D_a(g)]^*_{jj}[D_b(g)]_{ll} = \sum_{j, l}\delta_{ab}\delta_{jl}\delta_{jl}$$That is:
$$\frac{n_a}{N}\sum_{g \in G} \chi D_a(g)^*\chi D_b(g) = n_a\delta_{ab}$$or
$$\frac{1}{N}\sum_{g \in G} \chi D_a(g)^*\chi D_b(g) = \delta_{ab}$$This means character functions of $g$ in different irreducible representations are orthogonal.
3. Conjugacy Classes
A conjugacy class is defined as:
$$S_{g_\alpha} = \{g g_1 g^{-1} : g \in G\}$$This leads to $\chi D(g g_1 g^{-1}) = \chi D(g_1)$
Hence,
$$\frac{1}{N}\sum_{\alpha}k_\alpha \chi D_a(g_\alpha)^*\chi D_b(g_\alpha) = \delta_{ab}$$where $k_\alpha = n(S_{g_\alpha})$ is the number of elements in the conjugacy class.
From this, one can consider a new group $G_s$ classified by conjugacy classes, and define an orthonormal basis $|g_{\alpha}\rangle$ for this group. If a function $F(g_{\alpha})$ can be expressed as a linear combination of $\chi D_a(g_{\alpha})$, then it follows that the number of conjugacy classes is equal to the number of irreducible representations.
4. Functions Constant on Conjugacy Classes
For a function that is constant on a conjugacy class:
$$F(g_1) = \sum_{a,j,l} C^a_{jl} [D_a(g_1)]_{jl}$$We have:
$$F(g_1) = \frac{1}{N}\sum_{g} F(g g_1 g^{-1})$$ $$ = \frac{1}{N}\sum_{g,a,j,k,l,m} C^a_{jl} [D_a(g)]_{jk} [D_a(g_1)]_{km} [D_a(g)]^*_{lm}$$By the orthogonal relation:
$$F(g_1) = \sum_{a,j,k,l,m} \frac{1}{n_a}C^a_{jl} [D_a(g_1)]_{km} \delta_{jl} \delta_{km}$$Thus:
$$F(g_1) = \sum_{a,j,k} \frac{1}{n_a}C^a_{jj} [D_a(g_1)]_{kk}$$ $$ = \sum_{a} \frac{1}{n_a}\left(\sum_j C^a_{jj}\right) \chi D_a(g_1)$$Hence, $\chi D_a(g)$ forms a complete orthogonal basis for functions that are constant on a conjugacy class.
This has the consequence: number of conjugacy classes $=$ number of irreps.
5. Orthogonal Relations with Characters
From:
$$\frac{1}{N}\sum_{g \in G} \chi D_a(g)^*\chi D_b(g) = \delta_{ab}$$We have:
$$\sum_{\alpha} \frac{k_\alpha}{N}\chi D_a(g_\alpha)^*\chi D_b(g_\alpha) = \delta_{ab}$$We can define a square matrix $V$ with elements:
$$V_{a \alpha} = \sqrt{\frac{k_\alpha}{N}}\chi D_a(g_\alpha)$$Hence, from:
$$\sum_{\alpha}\frac{k_\alpha}{N} \chi D_a(g_\alpha)^*\chi D_a(g_\alpha) = 1$$or $VV^{\dagger} = 1$, or $V$ is a unitary matrix. The orthogonal relation:
$$\frac{k_\alpha}{N}\sum_{\alpha} \chi D_a(g_\alpha)^*\chi D_b(g_\alpha) = \delta_{ab}$$is for the column vectors of $V$, which means there exists from $V^{\dagger}V = 1$, another orthogonal relation with the row vectors of $V$:
$$\frac{k_\alpha}{N}\sum_{a} \chi D_a(g_\alpha)^*\chi D_a(g_\beta) = \delta_{\alpha\beta}$$6. Dimension Formula
Since $\sqrt{\frac{n_a}{N}}[D_a(g)]_{jk}$ forms an orthonormal basis, irrep $D_a(g)$ appears $n_a$ times in the completely reducible form of regular representation. And from:
$$\frac{1}{N}\sum_{g \in G} \chi D_a(g)^*\chi D_b(g) = \delta_{ab}$$We get:
$$n_a = \frac{1}{N}\sum_{g \in G} \chi D_a(g)^*\chi D(g)$$7. Projector $P_a$
From:
$$\frac{n_a}{N}\sum_{g \in G} [D_a(g)]^*_{jk}[D_b(g)]_{lm} = \delta_{ab}\delta_{jl}\delta_{km}$$We can derive:
$$\frac{n_a}{N}\sum_{g \in G} [D_a(g)]^*_{jj}[D_b(g)]_{lm} = \delta_{ab}\delta_{jl}\delta_{jm}$$ $$\frac{n_a}{N}\sum_{g,j} [D_a(g)]^*_{jj}[D_b(g)]_{lm} = \sum_{j} \delta_{ab}\delta_{jl}\delta_{jm}$$ $$\frac{n_a}{N}\sum_{g \in G} \chi D_a(g)^*[D_b(g)]_{lm} = \delta_{ab}\delta_{lm}$$This means the projector can be defined as:
$$P_a = \frac{n_a}{N}\sum_{g \in G} \chi D_a(g)^*D(g)$$