Character theory

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Basic definitions

In mathematics, the character of a group representation

ρ : GGLn

of a group (mathematics) G 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 character of an irreducible representation is called an irreducible character.

A special case of this kind of character occurs when ρ is a representation of degree 1; in this case, the character χ of ρ is called a linear character.

The kernel of a character χ is the set:

<math>\ker \chi := \left \lbrace g \in G \mid \chi(g) = \chi(1) \right \rbrace <math> where <math>\chi(1)<math> is the value of χ on the group identity.

If ρ is a representation of G of dimension k and 1 is the identity of G then

<math>\chi(1) = \operatorname{Tr}(\rho(1)) = \operatorname{Tr} \left ( \begin{vmatrix}1 & & 0\\ & \ddots & \\ 0 & & 1\end{vmatrix} \right ) = \sum_{i = 1}^k 1 = \dim \rho = k<math>

Unlike the situation with the character group, the characters of a group do not, in general, form a group themselves.

Character tables

The irreducible characters of a finite group form a character table which encodes many useful pieces of information about the group G in a compact form. Each row is labeled with a single irreducible character and contains the values of that character on each conjugacy class of G.

Here is the character table of C3, the cyclic group with three elements:

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

where u is a primitive third root of unity.

The character table is always square, and 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).

Orthogonality relations

One of the most important facts about the character table is that there are orthogonality relations on both the rows and the columns.

The inner product for characters (and hence for the rows of the character table) is given by:

<math>\left \langle \chi_i, \chi_j \right \rangle := \frac{1}{ \left | G \right | }\sum_{g \in G} \chi_i(g) \overline{\chi_j(g)}<math> where <math>\overline{\chi_j(g)}<math> means the complex conjugate of the value of <math>\chi_{j}<math> on g.

The orthogonality relation for columns is as follows:

For <math>g, h \in G<math> the sum <math>\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases}\left | C_G(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}<math>

where the sum is over all of the irreducible characters <math>\chi_i<math> of G.

The orthogonality relations can aid many computations including:

  • Decomposing an unknown character as a linear combination of irreducible characters,
  • Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
  • Finding the order of the group.

Character table properties

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.

Arithmetic with characters

Let <math>\rho<math> and <math>\sigma<math> be representations of G. Then the following identities hold:

<math>\Chi_{\rho \oplus \sigma} = \Chi_\rho + \Chi_\sigma<math>
<math>\Chi_{\rho \otimes \sigma} = \Chi_\rho \cdot \Chi_\sigma<math>
<math>\Chi_{\rho^*} = \overline {\Chi_\rho}<math>
<math>\Chi_{\textrm{Alt}^2 \rho}(g) = \frac{1}{2} \left[

\left(\Chi_\rho (g) \right)^2 - \Chi_\rho (g^2) \right]<math>

<math>\Chi_{\textrm{Sym}^2 \rho}(g) = \frac{1}{2} \left[

\left(\Chi_\rho (g) \right)^2 + \Chi_\rho (g^2) \right]<math>

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

References

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