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.

1 Definitions
2 Properties
- 2.1 Arithmetic properties
3 Character tables
- 3.1 Orthogonality relations
- 3.2 Character table properties
4 References

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Definitions

The character of a group representation

<math>\rho\ :\ G\ \to\ GL(n,\mathbb{C})</math>

of a group G is the function

<math>\chi_{\rho}\ :\ G\ \to\ \mathbb{C}</math>

which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). A character χ_ρ 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 χ_ρ is the set:

<math>\ker \chi_{\rho} := \left \lbrace g \in G \mid \chi_{\rho}(g) = \chi_{\rho}(1) \right \rbrace </math> where <math>\chi_{\rho}(1)</math> is the value of χ_ρ on the group identity.

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

<math>\chi_{\rho}(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.

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Properties

Characters are class functions, that is for all elements of a conjugacy class, χ is constant.

Isomorphic representations have the same characters. If the char(F)=0 then representations are isomorphic if and only if they have the same character.

If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' characters.

Every character <math>\chi\ (g) </math> is a sum of n m^th 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.

If F is algebraically closed and char(F) does not divide |G| then the number of irreducible characters of G is equal to the class number of G.

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Arithmetic properties

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 \rho = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho</math>.

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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).

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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.

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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.

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