Basic definitions

In mathematics, the character of a group representation Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

of a group G is the function In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

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. In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

Every character 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. In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

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:

where χ(1) 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

Unlike the situation with the character group, the characters of a group do not, in general, form a group themselves. In mathematics, a character group is the group of representations of a group by complex-valued functions. ...

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). In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identity mapping. ...

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. // Basic definitions In mathematics, the character of a group representation of a group G is the function which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). ...

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

where means the complex conjugate of the value of χ_j on g.

The orthogonality relation for columns is as follows:

For the sum

where the sum is over all of the irreducible characters χ_i 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 In fact if :, then there are irreducible characters χ₁...χ_n such that

.

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. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

See representation of a finite group for more details for the special case of finite groups. In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... In mathematics, a finite group is a group which has finitely many elements. ...

The characters of one-dimensional representations form a character group, which has important number theoretic connections. In mathematics, a character group is the group of representations of a group by complex-valued functions. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

Arithmetic with characters

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

where is the direct sum, is the tensor product, ρ ^* denotes the conjugate transpose of ρ, and Alt is the alternating product and Sym is the symmetric product, which is given by . In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ...

References

• Fulton, William; and Harris, Joe (1991). Representation Theory, A First Course, Springer, New York. ISBN 0-387-97495-4. See chapter 2.
• James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.), Cambridge University Press. ISBN 0-521-00392-X.
• 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.