GROUP THEORY TUTORIAL

Working with characters and irreducible representations

This tutorial explains the basic functionality of the Group Theory Package for working with characters and irreducible representations.

GTInstallGroup[group]Installation of a point group
GTCharacterTable[group]Calculation of the character table
Loading the Group Theory Package and installing the point group C3v.
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The character system of a given point group can be installed by using GTCharacterTable. With the help of an optional parameter, the character table can be printed by denotating the irreducible representations with an increasing index ("Bethe") or by the use of the Mulliken notation ("Mulliken"). The Bethe notation is used by default.
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GTAngularMomentumChars[classes,l]Calculation of the characters of the angular momentum representation.
GTIrep[characters,character table]Calculates, how often an irreducible representation occurs within a reducible representation.
GTGetIreps[group]Calculates the representation matrices and the character table.
To calculate the energy level splitting for atoms or molecules, the character of the (2l+1)-dimensional representation matrices of the angular momentum operator are needed. Those can be calculated by using GTAngularMomentumChars.
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To reduce a faithful representation and to calculate the number of times, an irreducible representation occurs within a reducible representation, GTIrep can be used.
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All representation matrices together with the character table can be calculated using GTGetIreps.
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GTDirectProductChars[characters1,characters2]Calculates the characters of a direct product representation from two given character systems.
For unitary representations Γ1 and Γ2 the characters of the direct product presentation Γ1⊗ are given by χ(T) = χ1(T)χ2(T). If two character systems are known, GTDirectProductRepresentation can be used, to calculate the character system of the direct product representation.
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