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What is a group character

  1. Jul 23, 2014 #1
    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).

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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