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Introduction to the accompanying software Ten[chi]ar
International Tables for Crystallography (2013). Vol. D, Section 1.2.7, pp. 62-70 [ doi:10.1107/97809553602060000901 ]
... in complicated cases errors can be made. This is therefore a situation in which it is best to rely on computer ... is accompanied by two software packages. Here we shall give a short introduction to the Ten[chi]ar package that deals ... of point groups in the second module. The latter play a role when determining the number of independent elements of ...
Determination of tensor products and their decomposition
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.5, p. 68 [ doi:10.1107/97809553602060000901 ]
... decomposition 1.2.7.4.5. Determination of tensor products and their decomposition Given a character (for an irreducible representation from the character table, or ... integers satisfying . (2) For each such decomposition there is a term where , and p is the number of nonzero integers ... values (). (4) The sign of the term is for a symmetrized power and for an antisymmetrized power. (5) The ...
Determination of the vector representation
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.4, p. 68 [ doi:10.1107/97809553602060000901 ]
... the vector representation 1.2.7.4.4. Determination of the vector representation For a point group G, its isomorphism class and its character table are known. For each conjugacy class, a representative element is given as word where the 's correspond ... the letters by the generating matrices, one obtains as product a matrix for which the trace is the character of ...
Diagonalization of the action matrix and determination of the invariant tensor
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.3, pp. 67-68 [ doi:10.1107/97809553602060000901 ]
... element of the tensor space under the group G is a vector v that is left invariant under each generator: If ... are equal to zero. The corresponding rows of Q form a basis for the solution space. (See example further on.) References ...
Action of the generators of the point group G on the basis
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.2, p. 67 [ doi:10.1107/97809553602060000901 ]
... in principle non-commutative. This polynomial can be written as a sum of the monomials in the basis taking into account ... anti)symmetry of and . In this way, basis element (a monomial) is transformed to To each generator of G corresponds ... previous equation changes to The function Det(g) is just a one-dimensional representation of the group G. The determinant ...
Construction of a basis
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.1, p. 67 [ doi:10.1107/97809553602060000901 ]
Construction of a basis 1.2.7.4.1. Construction of a basis As a basis for a tensor space without permutation symmetry, one ...
Algorithms
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4, pp. 67-70 [ doi:10.1107/97809553602060000901 ]
Algorithms 1.2.7.4. Algorithms 1.2.7.4.1. Construction of a basis | | As a basis for a tensor space without permutation symmetry, one may choose one ...
Characters
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.3, pp. 66-67 [ doi:10.1107/97809553602060000901 ]
... is selected in the main window by clicking `character'. A selection window opens in which a point group may be selected just as in the tensor ... organized according to dimension and geometric crystal class. Selection of a point group leads to the display of the character ...
Tensors
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.2, pp. 64-66 [ doi:10.1107/97809553602060000901 ]
... relations between the elements of tensors and pseudotensors invariant under a chosen point group and with specified permutation symmetry of the indices. Although the list of point groups provided in a database is limited to dimensions two and three, the program ... smaller than or equal to four. This is also not a restriction of the program, which works for arbitrary rank. ...
Overview
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.1, pp. 62-64 [ doi:10.1107/97809553602060000901 ]
... in complicated cases errors can be made. This is therefore a situation in which it is best to rely on computer ... is accompanied by two software packages. Here we shall give a short introduction to the Ten[chi]ar package that deals ... of point groups in the second module. The latter play a role when determining the number of independent elements of ...
International Tables for Crystallography (2013). Vol. D, Section 1.2.7, pp. 62-70 [ doi:10.1107/97809553602060000901 ]
... in complicated cases errors can be made. This is therefore a situation in which it is best to rely on computer ... is accompanied by two software packages. Here we shall give a short introduction to the Ten[chi]ar package that deals ... of point groups in the second module. The latter play a role when determining the number of independent elements of ...
Determination of tensor products and their decomposition
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.5, p. 68 [ doi:10.1107/97809553602060000901 ]
... decomposition 1.2.7.4.5. Determination of tensor products and their decomposition Given a character (for an irreducible representation from the character table, or ... integers satisfying . (2) For each such decomposition there is a term where , and p is the number of nonzero integers ... values (). (4) The sign of the term is for a symmetrized power and for an antisymmetrized power. (5) The ...
Determination of the vector representation
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.4, p. 68 [ doi:10.1107/97809553602060000901 ]
... the vector representation 1.2.7.4.4. Determination of the vector representation For a point group G, its isomorphism class and its character table are known. For each conjugacy class, a representative element is given as word where the 's correspond ... the letters by the generating matrices, one obtains as product a matrix for which the trace is the character of ...
Diagonalization of the action matrix and determination of the invariant tensor
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.3, pp. 67-68 [ doi:10.1107/97809553602060000901 ]
... element of the tensor space under the group G is a vector v that is left invariant under each generator: If ... are equal to zero. The corresponding rows of Q form a basis for the solution space. (See example further on.) References ...
Action of the generators of the point group G on the basis
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.2, p. 67 [ doi:10.1107/97809553602060000901 ]
... in principle non-commutative. This polynomial can be written as a sum of the monomials in the basis taking into account ... anti)symmetry of and . In this way, basis element (a monomial) is transformed to To each generator of G corresponds ... previous equation changes to The function Det(g) is just a one-dimensional representation of the group G. The determinant ...
Construction of a basis
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.1, p. 67 [ doi:10.1107/97809553602060000901 ]
Construction of a basis 1.2.7.4.1. Construction of a basis As a basis for a tensor space without permutation symmetry, one ...
Algorithms
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4, pp. 67-70 [ doi:10.1107/97809553602060000901 ]
Algorithms 1.2.7.4. Algorithms 1.2.7.4.1. Construction of a basis | | As a basis for a tensor space without permutation symmetry, one may choose one ...
Characters
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.3, pp. 66-67 [ doi:10.1107/97809553602060000901 ]
... is selected in the main window by clicking `character'. A selection window opens in which a point group may be selected just as in the tensor ... organized according to dimension and geometric crystal class. Selection of a point group leads to the display of the character ...
Tensors
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.2, pp. 64-66 [ doi:10.1107/97809553602060000901 ]
... relations between the elements of tensors and pseudotensors invariant under a chosen point group and with specified permutation symmetry of the indices. Although the list of point groups provided in a database is limited to dimensions two and three, the program ... smaller than or equal to four. This is also not a restriction of the program, which works for arbitrary rank. ...
Overview
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.1, pp. 62-64 [ doi:10.1107/97809553602060000901 ]
... in complicated cases errors can be made. This is therefore a situation in which it is best to rely on computer ... is accompanied by two software packages. Here we shall give a short introduction to the Ten[chi]ar package that deals ... of point groups in the second module. The latter play a role when determining the number of independent elements of ...
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