Character Table -- from Wolfram MathWorld

A character table records the group characters of the irreducible representations of a finite group. A finite group has a finite number of conjugacy classes and a finite number of distinct irreducible representations. The group character of a group representation is constant on a conjugacy class. Hence, the values of the group characters can be written as an array, known as a character table. Typically, the rows are given by the irreducible representations and the columns are given by the conjugacy classes.

A character table often contains enough information to identify a given abstract group and distinguish it from others. However, there exist groups that are not isomorphic which nevertheless have the same character table, for example (the symmetry group of the square) and (the quaternion group).

For example, the symmetric group on three letters has three conjugacy classes, represented by the permutations , , and . It also has three irreducible representations; two are one-dimensional and the third is two-dimensional:

1. The trivial representation .

2. The alternating representation, given by the permutation signature of the permutation, .

3. The standard representation on $V={(z_1,z_2,z_3):sumz_i=0}$ with

(1)

Using the vector basis , of , the standard representation is described on by the matrices

$phi^~_3({2,1,3})$			(2)
$phi^~_3({2,3,1})$			(3)

and hence the group character of the first matrix is 0 and that of the second is . The group character of the identity element is always the dimension of the vector space. The matrix trace of the alternating group representation is just the permutation signature of the permutation. Consequently, the character table for is shown below.

	1	2	3
		(12)	(123)
trivial	1	1	1
alternating	1		1
standard	2	0

Chemists and physicists use a special convention for representing character tables which is applied especially to the symmetry groups of molecules. These include the 32 crystallographic point groups compatible with a lattice, as well as other finite and infinite symmetry groups. In the example above, the numbered regions contain the following contents (Cotton 1990, pp. 90-92).

1. The symbol used to represent the group in question (in this case ).

2. The conjugacy classes, indicated by number and symbol, where the sum of the coefficients gives the group order of the group.

3. Mulliken symbols, one for each irreducible representation.

4. An array of the group characters of the irreducible representation of the group, with one column for each conjugacy class, and one row for each irreducible representation.

5. Combinations of the symbols , , , , , and , the first three of which represent the coordinates , , and , and the last three of which stand for rotations about these axes. These are related to transformation properties and basis representations of the group.

6. All square and binary products of coordinates according to their transformation properties.

The character tables for several symmetry groups are reproduced below using this notation. For the cyclic groups, pairs of complex-conjugate one-dimensional group characters are displayed together in rows labeled .