International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.2, p. 67
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As a basis for a tensor space without permutation symmetry, one may choose one consisting of non-commutative monomials. It has elements, where d is the dimension and r is the rank. In two dimensions, these are for , , , , for and , , , , , , , for . Note that .
If there is permutation symmetry among the indices , only polynomials occur in the basis for which . Then . If there is antisymmetry among these indices, one has the condition and . Therefore, in two dimensions, the basis for tensors of type (1 3)2 is , , , , , and for those of type [1 3]2 it is , . These bases can be obtained from the general basis by elimination.