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# Character Table

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 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 the conjugacy classes.

A character table often contains enough information to identify a given abstract group and distinguish it from others. However, there exist nonisomorphic groups 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 signature of the permutation, .

3. The standard representation on with

 (1)

The standard representation can be described on via the matrices

 (2) (3)

and hence the group character of the first matrix is 0 and that of the second is . The group character of the identity is always the dimension of the vector space. The trace of the alternating representation is just the permutation symbol 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 so-called point groups, which are the 32 finite symmetry groups possible in a lattice. 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 many of the point groups are reproduced below using this notation.

 1
 1 1 1
 1 1 1
 1 1 1
 1 1 1
 1 1 1 1 1 1
 1 1 1 1 1
 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 2 0
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0
 1 1 1 1 1 1 1 2 0 2 0
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0 0 2 2 0 0
 1 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 2 0
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0
 1 1 1 1 1 1 1 2 0 2 0
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0 0 2 2 0 0
 ... 1 1 ... 1 1 1 ... 2 ... 0 2 ... 0 2 ... 0

Conjugacy Class, Group, Group Character, Group Representation, Irreducible Representation, Point Groups

Portions of this entry contributed by Todd Rowland

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## References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Character Table

## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Character Table." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharacterTable.html