Character theory

In mathematics, the character of a group representation

ρ : G → GL_n

is the function χ : G -> C which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). For example, the character of the representation given above is given by: χ(1) = 2, χ(u) = 1 + u, χ(u²) = 1 + u².

If g and h are members of G in the same conjugacy class, then χ(g) = χ(h) for any character; the values of a character therefore have to be specified only for the different conjugacy classes of G. Moreover, equivalent representations have the same characters. If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' characters.

The characters of all the irreducible representations of a finite group form a character table, with conjugacy classes of elements as the columns, and characters as the rows. Here is the character table of C₃:

The character table is always square, and the rows and columns are orthogonal with respect to inner products on C^m (see orthogonality relation), which allows one to compute character tables more easily. 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).

Certain properties of the group G can be deduced from its character table:

• The order of G is given by the sum of (χ(1))² over the characters in the table.
• G is abelian if and only if χ(1) = 1 for all characters in the table.
• G has a non-trivial normal subgroup (i.e. G is not a simple group) if and only if χ(1) = χ(g) for some non-trivial character χ in the table and some non-identity element g in G.

The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements (D₈) have the same character table.

See representation of a finite group for more details for the special case of finite groups.