[[Group character]]
# Character table

The **character table** $\chi^\alpha_{c}$ of a group is a square[^square] [[Unitary operator|unitary matrix]] where each column is labeled by [[Conjugation by an element|conjugacy class]] and each row by an [[Irrep]].
Let $\alpha = 1, \dots, m$ label irreps and $c = 1,\dots,m$ label conjugacy classes $C_{c}$.
Then $\chi^\alpha_{c} = \chi^\alpha(x)$ for all $x \in C_{c}$.


[^square]: Since [[The number of conjugacy classes equals the number of non-equivalent irreps of a group]].

| $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ | $\{ (0,0) \}$ | $\{ (1,1) \}$ | $\{ (1,0) \}$ | $\{ (0,1) \}$ |
| -------------------------------------- | -------------:| -------------:| -------------:| -------------:|
| $\chi^1$ (trivial)                     |           $1$ |           $1$ |           $1$ |           $1$ |
| $\chi^2$                               |           $1$ |          $-1$ |          $-1$ |          $-1$ |
| $\chi^3$                               |           $1$ |          $-1$ |          $-1$ |           $1$ |
| $\chi^4$                               |           $1$ |          $-1$ |           $1$ |          $-1$ |


Properties characterising the character table, and thereby useful for determining its entries, include the [[Square sum of irrep dimensions]] and the [[Orthonormality of irreducible characters]], which gives
$$
\begin{align*}
\sum_{c =1}^m \frac{n_{c}}{\abs G} \chi^\alpha_{c} \chi^\beta_{c} &= \delta_{\alpha\beta} \\
\sum_{\alpha =1}^m \chi^\alpha_{c} \chi^\beta_{c} &= \frac{\abs G}{n_{c}} \delta_{\alpha\beta}
\end{align*}
$$


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