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

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

Section Determination of tensor products and their decomposition

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd 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 [m_{\alpha}] given above.

Fully symmetrized or antisymmetrized tensor products have characters given by[\eqalign{ n=2: \chi^\pm(R) &={1 \over 2!}\big(\chi (R)^2\pm\chi (R^2)\big)\cr n=3: \chi^\pm(R) &={1 \over 3!}\big(\chi (R)^3\pm 3\chi (R^2)\chi (R) +2\chi (R^3)\big)\cr n=4: \chi^\pm(R) &= {1 \over 4!}\big(\chi (R)^4\pm 6\chi (R^2)\chi (R)^2+3\chi (R^2)^2 \cr &\quad +8\chi (R^3)\chi (R)\pm 6\chi (R^4)\big)\cr n=5: \chi^\pm(R) &= {1 \over 5!}\big(\chi (R)^5\pm 10\chi (R^2)\chi (R)^3+15\chi (R^2)^2\chi (R)\cr&\quad +20\chi (R^3)\chi (R)^2 \pm 20\chi (R^3)\chi (R^2)\cr& \quad \pm 30\chi (R^4)\chi (R) +24\chi(R^5)\big)\cr n=6: \chi^\pm(R) &= {1 \over 6!}\big(\chi (R)^6\pm 15\chi(R^2)\chi (R)^4+45\chi (R^2)^2\chi (R)^2 \cr &\quad +40\chi (R^3)^2 \pm 15\chi(R^2)^3+ 40\chi (R^3)\chi (R)^3\cr &\quad \pm 120 \chi (R^3)\chi (R^2)\chi (R) \pm 90 \chi (R^4)\chi (R)^2\cr &\quad +90\chi (R^4)\chi (R^2) +144\chi (R^5)\chi (R)\cr&\quad\pm120 \chi (R^6)\big). }]From this follows immediately the dimension of the subspaces of symmetric and antisymmetric tensors:[\eqalign{n&=2: {1 \over 2}(d^2\pm d)\cr n&=3: {1 \over 6}(d^3\pm 3d^2+2d)\cr n&=4:{1 \over 24}(d^4\pm 6d^3+11 d^2\pm 6d)\cr n&=5: {1 \over 120}(d^5\pm 10d^4+35d^3\pm 50 d^2+24d)\cr n&=6: {1 \over 720}(d^6\pm 15d^5+85d^4\pm 225 d^3+274 d^2\pm 120 d). }]

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

  • (1) If n is the rank, the first step is to determine all possible decompositions [n = \textstyle\sum\limits_{i=1}^{n} f_{i}]with non-negative integers [f_i] satisfying [f_i \leq f_{i-1}].

  • (2) For each such decomposition [m=1,\ldots n_{\rm tot}] there is a term [P_m = \prod\limits_{i=1}^{p} \pmatrix{ N_i \cr f_i }(f_i -1)!, ]where [N_1=n], [N_i=N_{i-1}-f_{i-1}] [(i\,\gt\,1)] and p is the number of nonzero integers [f_i].

  • (3) If there are equal values of [f_i] in the mth decomposition, [P_m] should be divided by [t!] for each t-tuple of equal values ([f_{k+1}=\ldots =f_{k+t}]).

  • (4) The sign of the term [P_m] is [+1] for a symmetrized power and [\textstyle\prod\limits_{i=1}^{p} (-1)^{(f_i -1)}]for an antisymmetrized power.

  • (5) The expression for the character of the (anti)symmetrized power then is [\chi^{\pm} (R) = (1/M!)\textstyle\sum\limits_{m=1}^{n_{\rm tot}}{\rm sign}_m P_m \textstyle\prod\limits_{i=1}^{p} \chi (R^{f_i}). ]

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