# Definition A *character table* lists the [[Irreducible Representation|irreducible representations]] of a [[group]] in the columns and the [[Conjugacy Class|conjugacy classes]] in the rows (or vice versa). It only lists [[Primitive Character|primitive characters]]. # Setting Up Character Tables While it is rare you will have to set up a character table manually (they are almost always given in books and journals), here are the rules for how to do it 1. The number of [[Irreducible Representation|irreducible representations]] is equal to the number of [[Conjugacy Class|classes]]. 1. You can find the number of classes easily from the classification of the symmetry operations of the group. Otherwise, you have to compute all possible conjugates for all [[group element|group elements]] using the [[group multiplication table]]. 2. The [[dimension|dimensionalities]] of the [[Irreducible Representation|irreducible representations]] are found from $\sum_{i} \ell_i^2 = h$ , where $\ell_i$ is the dimension of the irreducible representation $\Gamma_i$ and $h$ is the [[group order|order]] of the group. 3. There is always a whole row of 1s in the character table for the identity representation. 4. The first column of the character table is always the trace for the unit matrix representing the identity element or class. This character is *always* $\ell_i$. Thus, the first column of the table is also filled in. 5. For all representations other than the identity representation $\Gamma_1$, the following relation is satisfied: $\sum_k N_k \chi^{(\Gamma_i)}(\mathcal{C}_k) = 0.$ This follows from the [[Orthogonality Relations for Character]]. 6. The first of the [[Orthogonality Relations for Character#First Orthogonality Relation|orthogonality relations for character]] holds for the rows of the table: $\sum_{k} N_k \chi^{(\Gamma_j)}(\mathcal{C}_k) \left[\chi^{(\Gamma_{j'})}(\mathcal{C}_k)\right]^* = h \delta_{\Gamma_j, \Gamma_{j'}}.$ It can be used for orthogonality (different rows) or normalization (same rows). 7. The second of the [[Orthogonality Relations for Character#Second Orthogonality Relation|orthogonality relations for character]] holds for the columns of the table: $\sum_{\Gamma_j} N_k \chi^{(\Gamma_j)}(\mathcal{C}_k) \left[\chi^{(\Gamma_{j})}(\mathcal{C}_{k'})\right]^* = h \delta_{k k'}.$ It can be used for orthogonality (different columns) or normalization (same columns). 8. From the character of the identity class $\chi^{(\Gamma_i)}(\mathcal{C}_e) = \ell_i$ and points (7 and 8), we get $\sum_{\Gamma_i}\chi^{(\Gamma_i)}(\mathcal{C}_k)\ell_i = 0.$