Ten![]() Calculations with tensors and characters M. Ephraim, T. Janssen, A. Janner and A. Thiers |
Point group | m |
Vector character | 3 1 |
Determinant character | 1 -1 |
Product(1,2) | 3 -1 |
Decompose(3) | 1 2 |
The multiplicity of the trivial representation in the pseudovector representation which is the product of the vector representation and the determinant representation is one. Therefore, there is one free parameter.
Point group | 3 |
Vector representation | 3 0 0 |
Symmetrized square | 6 0 0 |
Decompose | 2 2 2 |
Physical representation of irrep 2 | 2 -1 -1 |
The metric tensor invariant under the three-dimensional group 3 has two free parameters. The six-dimensional space of symmetric rank-two tensors has a two-dimensional invariant subspace. The remaining four-dimensional space carries the irreducible representations 2 and 3 twice. This space is twice the physically irreducible representation 2+3.
Point group | 4 |
Vector representation | 3 1 -1 1 |
Symmetrized square | 6 0 2 0 |
Symmetrized square of the former | 21 1 5 1 |
Decompose | 7 4 6 4 |
The elastic tensor is a rank-four tensor with intrinsic symmetry ((1 2)(3 4)). It can be obtained by taking twice the symmetrized square.
Point group | 4 |
Vector representation | 3 1 -1 1 |
Antisymmetrized square | 3 1 -1 1 |
Decompose | 1 1 1 |
The vector product of two vectors corresponds to a rank-two tensor with intrinsic symmetry [1 2]. The number of free parameters if the symmetry group is 4 is equal to 1.
Point group | mm2 |
Vector representation | 3 -1 1 1 |
Determinant representation | 1 1 -1 -1 |
Power ![]() |
9 1 1 1 |
Product(former, determinant rep.) | 9 1 -1 -1 |
Decompose | 2 3 2 2 |
The multiplicity of the trivial representation in the decomposition being 2, the number of free parameters in a pseudotensor of rank two invariant under mm2 is 2.
1 | 2 | 3 | 4 | 5 | ![]() |
|
1 | 1 | 2 | 3 | 4 | 5 | |
2 | 2 | 1 | 3 | 5 | 4 | |
3 | 3 | 3 | 1+2+3 | 4+5 | 4+5 | |
4 | 4 | 5 | 4+5 | 1+3+4+5 | 2+3+4+5 | |
5 | 5 | 4 | 4+5 | 2+3+4+5 | 1+3+4+5 | |
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If the selection rules for the matrix element
are
given in the following table.
1 | 2 | 3 | 4 | 5 | ![]() |
|
1 | 0 | 0 | * | 0 | 0 | |
2 | 0 | 0 | * | 0 | 0 | |
3 | * | * | * | 0 | 0 | |
4 | 0 | 0 | 0 | * | * | |
5 | 0 | 0 | 0 | * | * | |
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