math lessons - 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).

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:

     (1)  (u)  (u²)
  1   1    1    1
  χ₁  1    u    u²
  χ₂  1    u²   u

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.

The characters of one-dimensional representations form a character group, which has important number theoretic connections.

Properties

Let $ρ$ and $σ$ be representations of G. Then the following identities hold:

$\Chi_{\rho \oplus \sigma} = \Chi_\rho + \Chi_\sigma$

$\Chi_{\rho \otimes \sigma} = \Chi_\rho \cdot \Chi_\sigma$

$\Chi_{\rho^*} = \overline {\Chi_\rho}$

$\Chi_{\textrm{Alt}^2 \rho}(g) = \frac{1}{2} \left[ \left(\Chi_\rho (g) \right)^2 - \Chi_\rho (g^2) \right]$

$\Chi_{\textrm{Sym}^2 \rho}(g) = \frac{1}{2} \left[ \left(\Chi_\rho (g) \right)^2 + \Chi_\rho (g^2) \right]$

where $\rho \oplus \sigma$ is the direct sum, $\rho \otimes \sigma$ is the tensor product, $ρ *$ denotes the conjugate transpose of $ρ$ , and Alt is the alternating product $\textrm{Alt}^2 \rho = \rho \wedge \rho$ and Sym is the symmetric product , which is given by $\rho \otimes \sigma = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho$ .