TenXar
 Calculations with tensors and characters

M. Ephraim, T. Janssen, A. Janner and A. Thiers

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Subsections

Character

Starts the part of the program for calculations with characters of finite three-dimensional point groups.

Point Group

Click on the button to open a menu for selecting a point group.

View Character Table

Opens a separate window with the character table of the selected point group. Each column corresponds to a conjugation class.
  1. Conjugation classes represented by elements expressed in terms of generators $a,b,\dots $.
  2. The number of elements in each conjugation class.
  3. The order of the elements in the class.
  4. The characters of the irreducible representations.
  5. The classes to which the $p$th power of an element of this class belongs. These numbers are periodic with as period the order of the elements of the class. The $p$th power corresponds to the $p$th row.
  6. The three-dimensional vector representation. Tensors can be obtained by taking (symmetrized or antisymmetrized) powers of the vector representation.

Accept character table

Presents in the character window the characters for the irreducible representations and the vector representation. They form the first lines in the list of characters.

Power n=

The exponent needed for the specification of a power of a character.

Add

Add two characters selected in the character window by clicking the buttons at the beginning of the line. The result is added as a new line in the character window.

Product

Provides the product of two selected characters as a new line in the character list.

Power

Provides the $n$th power of one character selected in the character window. $n$ is specified in the fill-in window.

Symmetrized Square

Calculates the character of the representation which is the symmetrized second power of a selected representation by means of:

\begin{displaymath} \chi (g)_{\rm s}^{2}  = {{1}\over{2}}\left( \chi(g)^2 + \chi(g^2 ) \right) . \end{displaymath}

Antisymmetrized Square

Calculates the character of the representation which is the antisymmetrized second power of a selected representation by means of:

\begin{displaymath} \chi (g)_{\rm a}^{2}  = {{1}\over{2}}\left( \chi(g)^2 - \chi(g^2 ) \right) . \end{displaymath}

Physical character

For an irreducible representation it is checked whether it is also a physically irreducible representation. The expression

\begin{displaymath}\sum_{g\in G} \chi(g^2 )/\vert G\vert  = P , \end{displaymath}

where $\vert G\vert$ is the order of the group $G$, yields either 1, 0 or $-$1. If $P=0$ the representation does not have a real character. The corresponding physical representation is the sum of the representation and its complex conjugate. The character is the sum of the selected character and its complex conjugate. If $P=1$ the representation is equivalent to a real representation. If $P=-1$ the character is real, but not the representation matrices. The character is the double of the selected character.

Decompose

Gives the multiplicities of the irreducible representations in the decomposition of the selected character.

Copy

Copy the content of the character window to the worksheet, where it can be handled further.

Reset character list

The selection buttons in front of the characters in the list in the character window are made inactive.

Close

Close the character window and return to the worksheet.
next up previous
Next: Examples Up: The Buttons Previous: Tensor