GROUP THEORY TUTORIAL

Installation of point groups

Within the Group Theory Package the 32 crystallographic point groups are implemented. This notebook explains, how to install them.

GTInstallGroup[group]Installs a crystallographic point group
GTWhichRepresentation[]Gives the currently used representation
GTChangeRepresentation[representation]Changes the representation
Loading the Group Theory Package
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The implemented point groups are given by the nonaxial groups C1, Ci, Cs, the axial groups Cn, Cnv, Cnh, Dn, Dnd, Dnh (n = 2,3,4,6), S4, the tetrahedral groups T, Th, Td as well as the octahedral groups O and Oh. The installation of these groups gives a faithful representation using three dimensional rotation matrices of the Lie group SO(3).
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If spin is taken into account, the double groups of the 32 crystallographic point groups are needed. Double groups can be installed by changing the standard representation to SU(2).
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The groups Cn, Cnv (n = 2,3,4,6) as well as C1 and Cs are important for two dimensional systems. Those groups can be installed using two dimensional rotation matrices by choosing O(2) as a standard representation.
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The Group Theory Package remembers the used representation (SO(3), SU(2) or SO(2)) while working and thus all implemented modules will work correctly. But, in some cases, when two groups with different representations are installed confusions may occur. Therefore, the actual representation can be checked by using GTWhichRepresentation.
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Additionally, the standard representation can be switched manually between SO(3), SU(2) or SO(2) at all time, using GTChangeRepresentation
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GTTableToGroup[list of elements, multiplication table]Installs an arbitrary group from its multiplication table
The user can also install an arbitrary group from its multiplication table, by using GTTableToGroup. Suppose, that g is the order of the group, then the Group Theory Package will create a faithful representation using g-dimensional permutation matrices.
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Again, after installation of the group, the standard representation changes automatically.
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To change to permutation matrices use "Permutation".
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