Group Characters: Definition and Applications

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In summary, group characters are a useful tool for determining the irrep content of a representation without having to calculate all of the representation matrices. They can also be used to find the reality of an irrep and have various orthogonality relations. The character of an element a is the trace of its representation matrix, and the character of the identity element is the dimension of the representation. The character of the inverse of an element is the complex conjugate. All elements in a conjugacy class have the same character value, and the character values for the irreps form a matrix. By using these characters, one can find the irrep content of a representation.
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Definition/Summary

The character of a group representation is the trace of its representation matrices.

Group characters are useful for finding the irrep content of a representation without working out the representation matrices in complete detail.

Every element in a conjugacy class has the same character, regardless of representation. The character values for the irreps form a m*m matrix, for m classes, and thus m irreps.

Equations

The character of element a is
[itex]\chi(a) = Tr\ D(a)[/itex]

The character of the identity element is the dimension of the representation:
[itex]\chi(e) = n(D)[/itex]

The character of the inverse of an element is the complex conjugate:
[itex]\chi(a^{-1}) = \chi(a)^*[/itex]

Since all elements a of a conjugacy class A have the same character value,
[itex]\chi(A) = \chi(a)[/itex]

The characters of the irreps have various orthogonality relations.

For irreps k and l:
[itex]\sum_A n_A \chi^{(k)}(A) \chi^{(l)}(A)^* = n \delta_{kl}[/itex]
where n is the order of the group and n(A) is the order of class A.

For classes A and B:
[itex]\sum_k \chi^{(k)}(A) \chi^{(k)}(B)^* = \frac{n}{n_A}\delta_{AB}[/itex]

One can thus find the irrep content of a representation:
[itex]n(k) = \frac{1}{n} \sum_A n_A \chi(A) \chi^{(k)}(A)^*[/itex]

Extended explanation

One can find an irrep's reality in a simple way using its character.
[itex]\frac{1}{n}\sum_a \chi(a^2)[/itex]
is 1 for a real irrep, -1 for a pseudoreal irrep, and 0 for a complex irrep.

n is the order of the group (its number of elements).

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Greg Bernhardt said:
The character of a group representation is the trace of its representation matrices.
This is only an example. A group character is any homomorphism from the group into a multiplicative group of a field, i.e. we have a mapping ##x \longmapsto \sigma(x)## with ##\sigma(xy)=\sigma(x)\sigma(y)##. [Algebra Vol. 1,1970, B.L. van der Waerden].
 

Related to Group Characters: Definition and Applications

What is a group character?

A group character is a mathematical concept used in the field of group theory. It is a function that associates a complex number with each element of a group, representing the behavior of that element under group operations.

What is the importance of group characters?

Group characters are important because they allow us to study the structure and properties of groups in a more abstract and general way. They provide a way to classify and compare different groups and understand their symmetries and transformations.

How are group characters calculated?

Group characters are calculated using the character table, which is a matrix that contains the characters of all the elements of a group. The character table is constructed using the group's multiplication table and the representation matrices of each element.

What are some applications of group characters?

Group characters have various applications in mathematics, physics, and chemistry. They are used in the study of symmetry, crystallography, quantum mechanics, and many other areas. They also have applications in cryptography and coding theory.

Can a group character uniquely determine a group?

In general, no, a group character cannot uniquely determine a group. However, for finite groups, two groups can have the same character table only if they are isomorphic, meaning they have the same structure and properties. Therefore, in some cases, group characters can help identify and classify groups.

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