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
The standard representation can be described on
via the matrices
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.
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
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
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.
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