Group Characters: Definition and Applications

[Greg Bernhardt]

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
$\chi(a) = Tr\ D(a)$

The character of the identity element is the dimension of the representation:
$\chi(e) = n(D)$

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

Since all elements a of a conjugacy class A have the same character value,
$\chi(A) = \chi(a)$

The characters of the irreps have various orthogonality relations.

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

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

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

Extended explanation

One can find an irrep's reality in a simple way using its character.
$\frac{1}{n}\sum_a \chi(a^2)$
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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[fresh_42]

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].