Character theory

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In mathematics, more specifically in group representation theory, the character of a group representation is a function which associates to each element of the group an element of the field of the representation space. The character encodes many important properties of the group and can thus be used to study the group.

Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results relying on character theory include the Burnside theorem, and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a Sylow 2 subgroup that is a generalized quaternion group.

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[edit] Definitions

Let V be a finite-dimensional vector space over a field F and let \rho\colon G\to\mathrm{GL}(V) be a representation of a group G on V. The character of ρ is the function \chi_{\rho}\colon G\to F given by

\chi_{\rho}(g) = \mathrm{Tr}(\rho(g))\,

where Tr is the trace.

A character χρ is called irreducible if ρ is an irreducible representation. It is called linear if the dimension of ρ is 1. The kernel of a character χρ is the set:

\ker \chi_{\rho} := \left \lbrace g \in G \mid \chi_{\rho}(g) = \chi_{\rho}(1) \right \rbrace

where χρ(1) is the value of χρ on the group identity. If ρ is a representation of G of dimension k and 1 is the identity of G then

\chi_{\rho}(1) = \operatorname{Tr}(\rho(1)) = \operatorname{Tr} \begin{bmatrix}1 & & 0\\ & \ddots & \\ 0 & & 1\end{bmatrix} = \sum_{i = 1}^k 1 = k = \dim \rho

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

[edit] Properties

  • If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' characters.
  • Every character \chi\ (g) is a sum of n mth roots of unity where n is the degree (ie, the dimension n of the vector space over which GL(n) acts) of the representation, and m is the order of g.

[edit] Arithmetic 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 square, which is determined by

\rho \otimes \rho = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho.

[edit] 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, because the number of irreducible representations is equal to the number of conjugacy classes. 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).

[edit] 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:

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

With respect to this inner product, the irreducible characters are orthonormal: \left \langle \chi_i, \chi_j \right \rangle  = \begin{cases} 0  & \mbox{ if } i \ne j, \\ 1 & \mbox{ if } i = j. \end{cases}

The orthogonality relation for columns is as follows:

For g, h \in G the sum \frac{1}{ \left | G \right | }\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases}1/\left | C_G(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}

where the sum is over all of the irreducible characters χi of G and the symbol \left | C_G(g) \right | denotes the size of the conjugacy class of g.

The orthogonality relations can aid many computations including:

  • Decomposing an unknown character as a linear combination of irreducible characters.
  • Constructing the complete character table when only some of the irreducible characters are known.
  • Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
  • Finding the order of the group.

[edit] 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 (D4) 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.

[edit] References