GroupTheory`
GroupTheory`

GTRegularRepresentation

GTRegularRepresentation[group]

gives the regular representation of a group.

Details and Options

  • Consider a finite group with elements ,..., . The matrix elements of the -dimensional representation matrix are given by
  • .
  • For the character of each representation matrix it holds,
  • .
  • Let be a group with irreducible representations ,...,. Each irreducible representation occurs times within the regular representation , where is the dimension of the irreducible representation .
  • The following option can be given:
  • GOFast GOFastValueSkips the input validation
  • See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 5.1.

Examples

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Basic Examples  (1)

First, load the package:

As an example, the regular representation of the point group is calculated.

Options  (2)

GOFast  (2)

The evaluation time can be decreased, using GOFastTrue.

If GOFastFalse is used, a check of the input will be performed.

Applications  (1)

The statement "each irreducible representation occurs times within the regular representation , where is the dimension of the irreducible representation " will be verified. As an example the point group is used.

Calculate the 48×48-dimensional regular representation.

Calculate the character table of the pont group :

Calculate the characters of each class for the regular representation:

Use GTIrep to estimate the number of times each irreducible representation occurs within the regular representation.