UDC 512.972

MATHEMATICS

P. K. RASHEVSKII

ON THE STRUCTURE OF TENSORS ADMITTING A GIVEN GROUP OF INVARIANCE

(Presented by Academician I. G. Petrovskii, 24 I 1967)

Tensors in the complex vector space \(E^n\), generally speaking relative tensors, are considered. The latter means that under a change of basis \(\{e_i\}\)

\[ e_{i'}=A_{i'}^{i}e_i \tag{1} \]

the law of transformation of the coordinates of the tensor is complicated by multiplication by \((\operatorname{Det}\|A_{i'}^{i}\|)^p\), where \(p\) is an integer (possibly negative), called the weight of the tensor. If \(p=0\), then the tensor is called absolute. A special role is played by the \(n\)-vectors \(\omega_{i_1\ldots i_n}\), \(\varepsilon_{i_1\ldots i_n}\) (skew-symmetric in all indices), where \(\omega\) has weight \(-1\) and has the same coordinates in every basis, \(\omega_{12\ldots n}=1\); \(\varepsilon\) is an absolute tensor and is determined up to a constant factor. Analogously one defines \(\omega^{i_1\ldots i_n}\), \(\varepsilon^{i_1\ldots i_n}\) (in abbreviated notation \(\overline{\omega}, \overline{\varepsilon}\)).

The group of invariance of a given tensor (or finite system of tensors) \(Z\) is the subgroup \(\mathfrak{G}\subset GL(n,C)\) of all nonsingular matrices \(\|A_{i'}^{i}\|\) for which the transformations (1) of some initial basis \(\{e_i^0\}\) do not change the corresponding numerical values of the coordinates of \(Z\) (the dependence of \(\mathfrak{G}\) on the choice of the initial basis is trivial and is not reflected in the sense of the theory).

A concomitant \(K(Z)\) of a system of tensors \(Z\) is a tensor obtained from the tensors \(Z\) by the operations of addition, multiplication and contraction of tensors, multiplication of a tensor by a number, and permutation of the indices of a tensor (if all indices of the tensors \(Z\) are lower (upper), then the operation of contraction, of course, drops out).

Lemma 1. For every nonzero contravariant absolute tensor \(Z\) there exists a nonzero concomitant \(K(Z)\) which, in a suitable basis \(\{e_i\}\), has the form

\[ K(Z)=e_1\ldots e_1[e_1e_2]\ldots[e_1e_2][e_1e_2e_3]\ldots[e_1e_2e_3][e_1e_2\ldots e_n]\ldots[e_1e_2\ldots e_n]. \tag{2} \]

Here \(e_1\) is a vector, i.e. a unicontravariant tensor; \([e_1e_2]\) is a simple bivector, i.e. a twice contravariant tensor—the alternating product of the tensors \(e_1,e_2\); \([e_1e_2e_3]\) is a simple trivector, obtained analogously from \(e_1,e_2,e_3\), etc. These tensors, each taken some (possibly zero) number of times, are then multiplied among themselves.

Remark. If a nonzero contravariant absolute tensor \(Z\) has an irreducible group of invariance \(\mathfrak{G}\), then its concomitant (2) has, obviously, the simplified form

\[ K(Z)=[e_1e_2\ldots e_n]\ldots[e_1e_2\ldots e_n]. \tag{3} \]

We note that

\[ [e_1e_2\ldots e_n]^{i_1\ldots i_n}=\varepsilon^{i_1\ldots i_n}. \]

Lemma 1, with an inessential complication, is also true for relative contravariant tensors \(Z\).

All the results hold, of course, also for covariant tensors.

Lemma 2. Let a (relative) tensor \(W\) admit a group of invaria

of invariance \(\mathfrak G\) of a system of (relative) tensors \(Z\). Then there exist such concomitants \(S\) and \(T \ne 0\) of \(Z\), \(\omega\), \(\bar\omega\), that

\[ W\cdot T(Z,\omega,\bar\omega)=S(Z,\omega,\bar\omega) \tag{4} \]

(i.e., \(W\) is, as it were, the quotient of two concomitants).

In particular, if \(Z\) and \(W\) are covariant tensors, then

\[ W\cdot T(Z,\omega)=S(Z,\omega); \tag{5} \]

\(\bar\omega\) does not enter into the concomitants (and there is no contraction).

If \(Z\) and \(W\) are absolute tensors, then (4) takes the form:

\[ W\cdot T(Z,\delta)=S(Z,\delta), \tag{4'} \]

where \(T\) and \(S\) are concomitants of \(Z\) and of the invariant tensor

\[ \delta^i_j= \begin{cases} 0 & (i\ne j),\\ 1 & (i=j). \end{cases} \]

In particular, in the case of covariant \(Z, W\),

\[ W\cdot T(Z)=S(Z). \tag{5'} \]

Of course, for the case of contravariant \(Z, W\) analogous results hold.

Combining Lemma 1 (the remark) and Lemma 2, we obtain the following theorem.

Theorem. If the group of invariance \(\mathfrak G\) of a system of (relative) tensors \(Z\) is irreducible, then every (relative) tensor \(W\) admitting the group \(\mathfrak G\) has the form

\[ W=J^p\cdot S(Z,\omega,\bar\omega), \tag{6} \]

where \(J\) is a relative scalar of weight \(-1\) (one may take \(J=\varepsilon^{12\ldots n}\)); \(S\) is a concomitant; \(p\) is an integer; \(J^p\) necessarily admits the group \(\mathfrak G\).

In particular, if \(Z, W\) are absolute tensors:

\[ W=S(Z,\delta)\,\varepsilon\varepsilon\ldots\varepsilon \quad\text{or}\quad = S(Z,\delta)\,\bar\varepsilon\bar\varepsilon\ldots\bar\varepsilon, \tag{7} \]

where the product \(\varepsilon\varepsilon\ldots\varepsilon\) (or \(\bar\varepsilon\bar\varepsilon\ldots\bar\varepsilon\)) is contracted in all its indices with part of the indices of \(S(Z,\delta)\). This product may also be absent (the number of factors \(\varepsilon\) is equal to 0). If it is present, it must admit the group \(\mathfrak G\).

Finally, if the absolute tensors \(Z, W\) are covariant, then (7) simplifies:

\[ W=S(Z)\,\bar\varepsilon\ldots\bar\varepsilon, \tag{8} \]

and analogously for contravariant \(Z, W\):

\[ W=S(Z)\,\varepsilon\ldots\varepsilon. \tag{9} \]

Thus, under the condition of irreducibility of \(\mathfrak G\), the coordinates of \(W\) are expressed through the coordinates of \(Z\) by polynomials, and not only by rational functions, as in the general case (Lemma 2).

Example. All covariant absolute tensors in a complex simple Lie algebra, invariant under all its automorphisms, are concomitants of its covariant metric tensor \(g_{ij}(=C_{jp}^{i}C_{iq}^{p}\), where \(C_{jk}^{i}\) is the structure tensor) and of the covariant structure tensor \(C_{ijk}(=g_{ip}C_{jk}^{p})\), possibly also contracted with \(\bar\varepsilon\ldots\bar\varepsilon\) over all indices of this product. The number of factors \(\bar\varepsilon\) is arbitrary if all automorphisms have \(\operatorname{Det}=1\), and even if \(\operatorname{Det}=\pm1\).

Moscow State University
named after M. V. Lomonosov

Received
7 I 1967