TenXar
 Calculations with tensors and characters

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

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Subsections

Tensor

Starts the part of the program for determining invariant tensors.

Dimension

Type the number of dimensions in the open fill-in window.

Rank

Type the rank of the tensor in the open fill-in window.

Point Group

The name of the point group with which one wants to work can be given in the open fill-in window, or selected from a list. Clicking on the button Point Group opens a list of the seven systems in three dimensions or of the four systems in two dimensions. Clicking on one of the selection buttons opens a window with the symbols of the point groups in the selected system. The difference from the international notation is a - symbol in front of a digit, instead of a bar above it. By clicking on a button in the window a point group is selected.

View

A separate window is opened in which the generating matrices for the point group are shown as an object:
group(generator no. matrix$<$int$>$(matrix elements)
generator no. ...
).

Permutation Symmetry

The intrinsic or permutation symmetry can be typed into the open window or can be selected from a list for the lower dimensions (up to four).

The format is as follows: the indices of the tensor are numbered from 0 to rank $-$ 1. They should be typed with a space between the numbers. The order of the numbers is free. A change in the order corresponds to a change in setting.

Indices that are symmetric in the tensor are surrounded by (round) parentheses, antisymmetric indices by square brackets.

0 1 2 or 2 0 1 or 2 1 0 denote an arbitary rank-three tensor without permutation symmetry of the indices.

(0 1) 2 is a rank-three tensor symmetric in the first two indices (T$_{ijk}$=T$_{jik}$), (2 0) 1 a rank-three tensor symmetric under exchange of the first and third index (T$_{ijk}$=T$_{kji}$), and [2 1] 0 a rank-three tensor antisymmetric in the last two indices (T$_{ijk}$= -T$_{ikj}$).

Multiple symmetrizations are allowed: ((0 1)(2 3)) is a tensor of rank four invariant under permutation of first and second, third and fourth, and first and second pair of indices (T$_{ijkm}$=T$_{jikm}$=T$_{ijmk}$=T$_{kmij}$).

For low-rank tensors a list of preselected symmetries appears in a window opened by clicking on Perm. Symmetry, from which a specific choice can be made by clicking on one of the buttons.

Basis Transformation

The invariant tensors are calculated on the standard basis for the point group. For a different setting a basis transformation is applied by clicking on Basis Transformation and selecting a transformation. Even for the case of the standard setting one has to make a choice: No Transformation.

Tensor

Starts the calculation of the tensor of the given rank, invariant under the chosen point group and under the chosen permutations of the indices. The result appears in the worksheet.

Pseudo tensor

Starts the calculation of the pseudotensor of the given rank, invariant under the rotations of the chosen point group and under the chosen permutations of the indices, and obtaining an additional minus sign for the elements with determinant $-$1 in the point group. The result appears in the worksheet.

Close

Closes the window Tensor. This can be reopened by clicking on the top line button Tensor.
next up previous
Next: Character Up: The Buttons Previous: Delete