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

International Tables for Crystallography (2013). Vol. D, ch. 1.2, p. 67

Section Action of the generators of the point group G on the basis

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd Action of the generators of the point group G on the basis

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The transformation of the monomial [x_{i}x_{j}\ldots] under the matrix [g\in G] is given by the polynomial [\left [\textstyle\sum\limits_{m=1}^{d} g_{im}x_{m} \right] \times \left [\textstyle\sum\limits_{n=1}^{d} g_{jn}x_{n} \right] \ldots ,]which is in principle non-commutative. This polynomial can be written as a sum of the monomials in the basis taking into account the eventual (anti)symmetry of [xy] and [yx]. In this way, basis element (a monomial) [e_{i}] is transformed to [ge_i=\textstyle\sum\limits_{j=1}^dM(g)_{ji}e_j.]To each generator of G corresponds such an action matrix M.

The action matrix changes if one considers pseudotensors. In the case of pseudotensors, the previous equation changes to [ge_i={\rm Det}(g)\textstyle\sum\limits_{j=1}^dM(g)_{ji}e_j.]The function Det(g) is just a one-dimensional representation of the group G. The determinant is either [+1] or [-1].

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