GROUP THEORY SYMBOL

GTTableToGroup

GTTableToGroup[list of elements,multiplication table]
gives a faithful representation of an arbitrary group from a given list of elements and a multiplication table, using permutation matrices.

DetailsDetails

  • Suppose a group with order , then GTTableToGroup gives a faithful representation using -dimensional permutation matrices.
  • GTTableToGroup changes the standard representation to "permutation matrices". (See GTWhichRepresentation)
  • The following option can be given:
  • GOVerboseTrueControls the output of additional information
  • See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 3.1

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

First, load the package:

In[1]:=
Click for copyable input

Install Klein's four group from a multiplication table

In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=

We can construct also another example. First, the group C4v is installed and the multiplication table is calculated.

In[4]:=
Click for copyable input

Now we perform an isomorphic mapping to a set of other elements.

In[5]:=
Click for copyable input
Out[5]=

A new multiplication table defined in the new symbols was created. Now we install the group from the multiplication table.

In[6]:=
Click for copyable input

Note, during the installation the standard representation is switched to "Permutation", because the group elements are installed in terms of permutation matrices.

The first matrices are given by:

In[7]:=
Click for copyable input
Out[7]=

After the installation of the group we can calculate the multiplication table again.

In[8]:=
Click for copyable input

Now we can apply other commands to this special representation of C4v.

In[9]:=
Click for copyable input
Out[9]=
In[10]:=
Click for copyable input
Out[10]=

Note, it might be that not all commands of GTPack can handle the permutation representation in the moment.

We swithc back to the standard representation of rotation matrices.

In[11]:=
Click for copyable input

Options  (2)Options  (2)

Top