Construction of a basis

Ephra[iuml ]m, M.; Janssen, T.; Janner, A.; Thiers, A.

doi:10.1107/97809553602060000629

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.2, p. 67

Section 1.2.7.4.1. Construction of a basis

M. Ephraïm,^b T. Janssen,^a A. Janner^c and A. Thiers^d

1.2.7.4.1. Construction of a basis

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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 [d^r] elements, where d is the dimension and r is the rank. In two dimensions, these are [x,y] for [r=1] , [xx] , [xy] , [yx] , [yy] for [r=2] and [xxx] , [xxy] , [xyx] , [xyy] , [yxx] , [yxy] , [yyx] , [yyy] for [r=3] . Note that $[xy\neq yx]$ .

If there is permutation symmetry among the indices $[i_{1},\ldots, i_{p}]$ , only polynomials $[x_{i_1}x_{i_2}\ldots x_{r}]$ occur in the basis for which $[i_{1}\leq i_{2} \leq \ldots \leq i_{p}]$ . Then $[x_{i_1}x_{i_2}=x_{i_2}x_{i_1}]$ . If there is antisymmetry among these indices, one has the condition $[i_{1} \,\lt\, i_{2} \,\lt\, \ldots \,\lt\, i_{p}]$ and $[x_{i_1}x_{i_2}=-x_{i_2}x_{i_1}]$ . Therefore, in two dimensions, the basis for tensors of type (1 3)2 is [xxx] , [xxy] , [xyx] , [xyy] , [yxy] , [yyy] and for those of type [1 3]2 it is [xxy] , [xyy] . These bases can be obtained from the general basis by elimination.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, p. 67