Consequences of the grand orthogonality theorem ↩

shorthands: {}
aliases: [Consequences of the Schur orthogonality theorem, Consequences of the Schur orthogonality relations]
created: 2021-10-28 22:18:10
modified: 2022-01-10 06:09:05

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Where denotes the number of inequivalent irreducible representations of . The most important observation is that is limited by something.
Actually, the equality holds as well, the proof can be seen here.

C2

For characters of irreducible representations:

(According to Maschke's theorem says that if one can "sum over" the group elements, then all of its finite dimensional representations are unitary-equivalent.)

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The characters of irreducible representations are unit vectors in .¹

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For two irreducible representations of a given

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Another limit on the number of inequivalent irreducible representations: since characters of inequivalent irreducible representations are perpendicular to each other, their numbers cannot exceed the number of conjugacy classes in the group, that is:

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For any fully decomposable or irreducible representation, , the scalar product² is always a natural number, whose meaning is: "by what multiplicity does contain ?".
Then the representation can be written as the direct product of the group's irreducible representations like this:

Where is the multiplicity of the irreducible representation.

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For two fully decomposable or irreducible representations of a given .

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A representation is irreducible if and only if , that is its character is a unit vector in .³

Examples

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Examples for some point groups:

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point group

Irreducible representations of the point group:

The characters () are equal to the representation because .
Their products:

point group

The characters of irreducible representations of the point group:

Their scalar products: