International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.2, p. 68

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)