GroupTheory`

GTCharacterTable

gives the character table of a group.

Details and Options

The trace of the representation matrix of an element of a group is called character . The character systems of the irreducible representations of a finite group are conventionally displayed in the form of a "character table".
All elements in the same class have the same character.
The number of inequivalent irreducible representations is equal to the number of classes of a finite group .
For a finite group a necessary and sufficient condition for two representations to be equivalent is provided by the equality of their character systems.
If a group is finite, the sum of the squares of the dimensions of the inequivalent irreducible representations is equal to the order of .
The output of GTCharacterTable is a list of all classes, a list of the character system and a list of the names of the irreducible representations.
Attention: The names of the irreducible representations are generated automatically and may differ from tables in the literature!
The following options can be given:

GOIrepNotation	"Bethe"	Notation for irreducible representations
GOVerbose	True	Controls the output of information
GOFast	GOFastValue	Skips the input vlidation
GOReality	False	Provides information about the reality of the irreducible representations
GOMethod	"NumericalApproximant"	Method for internal eigenvalue solution

See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 5.8

Examples

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Basic Examples (1)

First, load the package:

As an example, we calculate the character table of the point group C_3v.

Options (9)

GOFast (2)

The evaluation time can be decreased, using GOFastTrue.

If GOFastFalse is used, a check of the input will be performed.

GOIrepNotation (4)

GTCharacterTable can estimate automatically the notation according to Mulliken (The Journal of Chemical Physics, AIP, 23, 1997-2011, (1953)) .

With Bethe notation all irreducible representations are labelled with an increasing index.

GTCharacterTable can estimate automatically the notation according to Bouckaert (L. Bouckaert, R. Smoluchowski and E. Wigner, Phys. Rev., APS, 50, 58-67 (1936)) .

GTCharacterTable also allows for manual input.

GOVerbose (1)

Using GOVerboseFalse the printed output can be suppressed.

GOReality (1)

Using GORealityTrue additional information about the reality of irreducible representations is printed.

GOMethod (1)

The character table is obtained by solving an eigenvalue problem simultaneously diagonalizing several matrices. For that a random number method is used which can be solved numerically or analytically.

"NumericalApproximant" and "Analytic" should give the same results. For "NumericalApproximant", the Eigenvalue problem is solved numerically (fast for large groups) and transformed in to radicals using RootApproximant and ToRadicals (sometimes slow for complicated root expressions).