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

\rho

and

\sigma

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 (g) \right)^2 - \Chi_\rho (g^2) \right

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

where

\rho \oplus \sigma

is the direct sum,

\rho \otimes \sigma

is the tensor product,

\rho^*

denotes the conjugate transpose of

\rho

, 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

References

William Fulton, Joe Harris, Representation Theory, A First Course, (1991) Springer, New York. ISBN 0-387-97495-4 See chapter 2.
http://planetmath.org/encyclopedia/Character.html
Character Tables for chemically important point groups - Lists most of the point groups and gives their character tables in notation used in Chemistry.

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