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

International Tables for Crystallography (2013). Vol. D, ch. 1.2, p. 68

## Section 1.2.7.4.5. Determination of tensor products and their decomposition

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd

#### 1.2.7.4.5. Determination of tensor products and their decomposition

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Given a character (for an irreducible representation from the character table, or for the vector representation, for example), the character of the standard rank n tensor is the nth power of the character and can be decomposed with the multiplicity formula for given above.

Fully symmetrized or antisymmetrized tensor products have characters given byFrom this follows immediately the dimension of the subspaces of symmetric and antisymmetric tensors:

The general expression for arbitrary rank can be determined as follows. (See also Section 1.2.2.7)

 (1) If n is the rank, the first step is to determine all possible decompositions with non-negative integers satisfying . (2) For each such decomposition there is a term where , and p is the number of nonzero integers . (3) If there are equal values of in the mth decomposition, should be divided by for each t-tuple of equal values (). (4) The sign of the term is for a symmetrized power and for an antisymmetrized power. (5) The expression for the character of the (anti)symmetrized power then is