Tensor representations

Janssen, T.

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, pp. 42-43

Section 1.2.2.7. Tensor representations

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.2.7. Tensor representations

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When $[V_{1},\ldots, V_{n}]$ are linear vector spaces, one may construct tensor products of these spaces. There are many examples in physics where this notion plays a role. Take the example of a particle with spin. The wave function of the particle has two components, one in the usual three-dimensional space and one in spin space. The proper way to describe this situation is via the tensor product. In normal space, a basis is formed by spherical harmonics $[Y_{lm}]$ , in spin space by the states $[|ss_{z}\rangle]$ . Spin–orbit interaction then plays in the [(2l+1)(2s+1)] -dimensional space with basis $[|lm\rangle\otimes |ss_{z}\rangle]$ . Another example is a physical tensor, e.g. the dielectric tensor $[\varepsilon_{ij}]$ of rank 2. It is a symmetric tensor that transforms under orthogonal transformations exactly like a symmetric bi-vector with components $[v_{i}w_{j}+v_{j}w_{i}]$ , where $[v_{i}]$ and $[w_{i}]$ ( [i=1,2,3] ) are the components of vectors $[{\bf v}]$ and $[{\bf w}]$ . A basis for the space of symmetric bi-vectors is given by the six vectors ( $[{\bf e}_{i}\otimes{\bf e}_{j}+{\bf e}_{j}\otimes{\bf e}_{i}]$ ) ( $[i\leq j]$ ). The space of symmetric rank 2 tensors has the same transformation properties .

A basis for the tensor space $[V_{1}\otimes V_{2}\otimes \ldots \otimes V_{n}]$ is given by $[{\bf e}_{1i}\otimes {\bf e}_{2j}\otimes \ldots \otimes {\bf e}_{nk}]$ , where $[i=1,2,\ldots, d_{1}]$ ; $[j=1,2,\ldots d_{2}]$ ; $[\ldots]$ ; $[k=1,2,\ldots, d_{n}]$ . Therefore the dimension of the tensor product is the product of the dimensions of the spaces $[V_{i}]$ (see also Section 1.1.3.1.2 ). The tensor space consists of all linear combinations with real or complex coefficients of the basis vectors. In the summation one has the multilinear property $[\left(\textstyle\sum\limits_{i=1}^{d_{1}} c_{1i}{\bf e}_{1i} \right) \otimes \left(\textstyle\sum\limits_{j=1}^{d_{2}} c_{2j}{\bf e}_{2j} \right) \otimes \ldots = \textstyle\sum\limits_{ij\ldots} c_{1i}c_{2j}\ldots {\bf e}_{1i}\otimes {\bf e}_{2j}\otimes \ldots .\eqno (1.2.2.40)]$ In many cases in practice, the spaces $[V_{i}]$ are all identical and then the dimension of the tensor product $[V^{\otimes n}]$ is simply $[d^{n}]$ .

The tensor product of n identical spaces carries in an obvious way a representation of the permutation group $[S_{n}]$ of n elements. A permutation of n elements is always the product of pair exchanges. The action of the permutation (12), that interchanges spaces 1 and 2, is given by $[P_{12}{\bf e}_{i}\otimes {\bf e}_{j}\otimes {\bf e}_{k}\otimes \ldots = {\bf e}_{j}\otimes {\bf e}_{i}\otimes {\bf e}_{k}\otimes \ldots .\eqno (1.2.2.41)]$ Two subspaces are then of particular interest, that of the tensors that are invariant under all elements of $[S_{n}]$ and those that get a minus sign under pair exchanges. These spaces are the spaces of fully symmetric and antisymmetric tensors, respectively.

If the spaces $[V_{1},\ldots, V_{n}]$ carry a representation of a finite group K, the tensor product space carries the product representation. $[\displaylines{{\bf e}_{1j_{1}}\otimes {\bf e}_{2j_{2}}\otimes \ldots\hfill\cr\quad = {\bigotimes\limits_{i=1}^{n}} {\bf e}_{ij_{i}} \rightarrow \textstyle\sum\limits_{k_{1}k_{2}\ldots} \Gamma_{1}(R)_{k_{1}j_{1}}\Gamma_{2}(R)_{k_{2}j_{2}}\ldots {\bf e}_{1k_{1}}\otimes{\bf e}_{2k_{2}}\otimes \ldots .\hfill\cr\hfill (1.2.2.42)}]$ The matrix $[\Gamma (R)]$ of the tensor representation is the tensor product of the matrices $[\Gamma_{i}(R)]$ . In general, this representation is reducible, even if the representations $[\Gamma_{i}]$ are irreducible. The special case of [n=2] has already been discussed in Section 1.2.2.3.

From the definition of the action of $[R\in K]$ on vectors in the tensor product space, it is easily seen that the character of R in the tensor product representation is the product of the characters of R in the representations $[\Gamma_{i}]$ : $[\chi (R) = \textstyle\prod\limits_{i=1}^{n} \chi_{i}(R). \eqno (1.2.2.43)]$ The reduction in irreducible components then occurs with the multiplicity formula. $[m_{\alpha} = ({1}/{N})\textstyle\sum\limits_{R\in K}\chi_{\alpha}^{*}(R)\textstyle\prod\limits_{i=1}^{n} \chi_{i}(R). \eqno (1.2.2.44)]$ If the tensor product representation is a real representation, the physically irreducible components can be found by first determining the complex irreducible components, and then combining with their complex conjugates the components that cannot be brought into real form.

The tensor product of the representation space V with itself has a basis $[{\bf e}_{i}\otimes{\bf e}_{j}]$ ( $[i,j=1,2,\ldots, d]$ ). The permutation (12) transforms this into $[{\bf e}_{j}\otimes{\bf e}_{i}]$ . This action of the permutation becomes diagonal if one takes as basis $[{\bf e}_{i}\otimes{\bf e}_{j}+{\bf e}_{j}\otimes{\bf e}_{i}]$ ( $[1\leq i \leq j \leq d]$ , spanning the space $[V_{s}^{\otimes 2}]$ ) and $[{\bf e}_{i}\otimes{\bf e}_{j}-{\bf e}_{j}\otimes{\bf e}_{i}]$ ( $[1\leq i \,\lt\, j \leq d]$ , spanning the space $[V_{a}^{\otimes 2}]$ ). If one considers the action of K, one has with respect to the first basis $[\chi (R)=\chi_{\alpha}(R)^{2}]$ if V carries the representation with character $[\chi_{\alpha}(K)]$ . With respect to the second basis, one sees that the character of the permutation [P=(12)] is given by $[{{1}\over{2}} d(d+1) -{{1}\over{2}} d(d-1)=d]$ . The action of the element $[R\in K]$ on the second basis is $[R\left({\bf e}_{i}\otimes{\bf e}_{j}\pm{\bf e}_{j}\otimes{\bf e}_{i} \right)=\textstyle\sum\limits_{kl}(\Gamma_{\alpha}\otimes \Gamma_{\alpha})(R)_{kl,ij} \left({\bf e}_{i}\otimes{\bf e}_{j}\pm{\bf e}_{j}\otimes{\bf e}_{i}\right). ]$ This implies that both $[V_{s}^{\otimes 2}]$ and $[V_{a}^{\otimes 2}]$ are invariant under R. The character in the subspace is $[\chi^{+}(R) = \textstyle\sum\limits_{k\leq l}(\Gamma_{\alpha}\otimes \Gamma_{\alpha})(R)_{kl,kl}\eqno(1.2.2.45)]$ for the symmetric subspace and $[ \chi^{-}(R) = \textstyle\sum\limits_{k \,\lt\, l}(\Gamma_{\alpha}\otimes \Gamma_{\alpha})(R)_{kl,kl} \eqno(1.2.2.46)]$ for the antisymmetric one. Consequently one has $[\chi^{\pm}(R) = {\textstyle{{1}\over{2}}} (\chi_{\alpha}(R)^{2}\pm\chi_{\alpha}(R^{2}));\quad d^{\pm} = {\textstyle{{1}\over{2}}} d_{\alpha}(d_{\alpha}\pm 1). \eqno (1.2.2.47)]$

For $[n\,\gt\,2]$ , the tensor product space does not carry just a symmetric and an antisymmetric subspace, but also higher-dimensional representations of the permutation group $[S_{n}]$ . The derivation of the character of the fully symmetric and fully antisymmetric subspaces remains rather similar. The formulae for the character of the representation of K carried by the fully symmetric ( [+] ) and fully antisymmetric (−) subspace, respectively, for [n=1,2,3,4,5,6] are $[\eqalign{n=2: \chi^\pm(R) &= {{1}\over{2!}}\big(\chi (R)^2\pm\chi (R^2)\big)\cr n=3: \chi^\pm(R) &= {{1}\over{3!}}\big(\chi (R)^3\pm 3\chi (R^2)\chi (R) +2\chi (R^3)\big)\cr n=4: \chi^\pm(R) &= {{1}\over{4!}}\big(\chi (R)^4\pm 6\chi (R^2)\chi (R)^2+3\chi (R^2)^2 \cr &\quad +8\chi (R^3)\chi (R)\pm 6\chi (R^4)\big)\cr n=5: \chi^\pm(R) &= {{1}\over{5!}}\big(\chi (R)^5\pm 10\chi (R^2)\chi (R)^3+15\chi (R^2)^2\chi (R) \cr&\quad +20\chi (R^3)\chi (R)^2 \pm 20\chi (R^3)\chi (R^2)\cr &\quad \pm 30 \chi (R^4)\chi (R) +24\chi(R^5)\big)\cr n=6: \chi^\pm(R) &= {{1}\over{6!}}\big(\chi (R)^6\pm 15\chi(R^2)\chi (R)^4+45\chi (R^2)^2\chi(R)^2\cr&\quad +40\chi (R^3)^2 \pm 15\chi(R^2)^3 +40\chi (R^3)\chi (R)^3\cr&\quad \pm 120 \chi (R^3)\chi (R^2)\chi (R) \pm 90 \chi (R^4)\chi (R)^2\cr&\quad+90\chi (R^4)\chi (R^2) +144\chi (R^5)\chi (R)\cr&\quad\pm120 \chi (R^6)\big) \cr}]$ From this follows immediately the dimension of the two subspaces: $[\eqalign{n&=2\!: {\textstyle{1}\over{2}} (d^2\pm d)\cr n& =3\!: {\textstyle{1}\over{6}}(d^3\pm 3d^2+2d)\cr n&=4\!: {\textstyle{1}\over{24}}(d^4\pm 6d^3+11 d^2\pm 6d)\cr n& =5\!: {\textstyle{1}\over{120}}(d^5\pm 10d^4+35d^3\pm 50 d^2+24d)\cr n& =6\!: {\textstyle{1}\over{720}}(d^6\pm 15d^5+85d^4\pm 225 d^3+274 d^2\pm 120 d).\cr}]$ These expressions are based on Young diagrams. The procedure will be exemplified for the case of [n=5] . In the expression for $[\chi^{\pm}]$ occur the partitions of n in groups of integers: [Scheme scheme1] [5=] [4+1=] [3+2=] [3+1+1=] [2+2+1=] [2+1+1+1=] [1+1+1+1+1] . Each partition corresponds with a Young diagram with as many rows as there are terms in the sum, and in each row the corresponding number of boxes. The total number of boxes is n. Each partition corresponds with a term $[\chi (R^{i_{1}})\chi (R^{i_{2}})\ldots]$ such that $[\textstyle\sum_{j}i_{j}=n]$ . Here [i_1] is the number of boxes in the first row etc. The prefactor then is the number of possible permutations compatible with the partition. For example, the partition [2+2+1] allows the permutations $[\displaylines{(12)(34)(5) \;\, (13)(24)(5)\;\, (14)(23)(5)\;\, (12)(35)(4)\;\, (13)(25)(4)\cr (15)(23)(4)\;\, (12)(45)(3)\;\, (14)(25)(3)\;\, (15)(24)(3)\;\, (13)(45)(2)\cr (14)(35)(2)\;\, (15)(34)(2)\;\, (23)(45)(1)\;\, (24)(35)(1)\;\, (25)(34)(1)}]$ The sign of all these permutations is even: they are the product of an even number of pair interchanges. The prefactor for the term $[\chi (R^{2})\chi (R^{2})\chi (R)]$ is then [+15/5!] .