Definition and Example
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on the representatives of the respective conjugacy class of G (because characters are class functions). The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree 1 are known as linear characters.
Here is the character table of C3 = , the cyclic group with three elements and generator u:
| (1) | (u) | (u2) | |
| 1 | 1 | 1 | 1 |
| χ1 | 1 | ω | ω2 |
| χ2 | 1 | ω2 | ω |
where ω is a primitive third root of unity.
The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes. The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1).
Read more about this topic: Character Table
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