Reports of the Academy of Sciences of the USSR
1964. Volume 154, No. 1

MATHEMATICS

N. I. GLAZUNOV

TENSOR FIELDS ON COMPACT HOMOGENEOUS SPACES ADMITTING A SPHERICAL MODEL

(Presented by Academician I. G. Petrovskii, 18 VII 1963)

Let there be given a homogeneous space \(K=G/H\), where \(G \supset H\) are Lie groups, \(G\) is connected, and \(H\) is its closed subgroup. Suppose that we have constructed a faithful representation of the group \(G\) in the form of a group of motions \(\varphi(G)\) of the unit sphere \(S_n\) of the Euclidean space \(E_{n+1}\), admitting such a closed surface of transitivity \(K^*\) that the stationary subgroup \(H\) of the group \(G\) is mapped into the stationary subgroup \(\varphi(H)\) of the group \(\varphi(G)\), considered as a group of transformations on \(K^*\), and at the same time the representation \(\overline{\varphi}(G)\) by a group of motions on \(K^*\) is faithful. In this case we shall say that \(K^*\) is a spherical model of the homogeneous space \(K\).

It is clear that, in order for a homogeneous space \(K\) to admit a spherical model, it is necessary that the group \(G\), and consequently also \(K\), be compact. Conversely, in view of the compactness of \(G\), it has a faithful representation by orthogonal matrices, and moreover the compact subgroup \(H\) is singled out algebraically in this representation; then one can construct a spherical model of the homogeneous space \(K\) in the same way as in \((^1)\) (p. 198) the projective model of the homogeneous space \(K\) is constructed.

In \((^2)\) a description was given of the spaces of tensor fields on the \(n\)-dimensional sphere \(S_n\), irreducible with respect to the group of motions \(O(n+1)\) (which may be regarded as the homogeneous space \(K=O(n+1)/O(n)\)).

The purpose of the present paper is, first, to give a scheme for obtaining analogous results in the general case, when \(K=G/H\) is a homogeneous space admitting a spherical model; and, second, to carry this out explicitly for the Grassmann manifold \(K=O(m)/O(m-k)O(k)\).

The induced tensor field \(kT_{\alpha_1,\ldots,\alpha_l}\), introduced in \((^2)\), may be interpreted as the projection onto the sphere \(S_n\) of a constant tensor field \(T_{i_1,\ldots,i_k}\) of the space \(E_{n+1}\), with the annihilated indices being projected onto the normal to \(S_n\), and the remaining ones onto the plane tangent to \(S_n\).

Let us take a spherical model \(K^*\) of our homogeneous space \(K\) on the unit sphere \(S_n\) of the Euclidean space \(E_{n+1}\). We decompose the set of all possible tensors in \(E_{n+1}\) into linear spaces \(\Pi\), irreducible with respect to the group \(\varphi(G)\). Just as in \((^2)\), we project, in all the indicated ways, these tensor spaces \(\Pi\), irreducible with respect to \(\varphi(G)\), onto the sphere \(S_n\). We obtain, as in \((^2)\), a complete system of spaces \(\Pi'\) of tensor fields on \(S_n\), irreducible with respect to the group \(\varphi(G)\) (complete in the sense that every continuous tensor field on \(S_n\) admits uniform approximation by finite sums of our induced tensor fields).

Now we project the tensor fields obtained on \(S_n\) onto our model \(K^*\). In doing so, we project all indices onto the plane tangent to \(K^*\). The tensor fields on \(K^*\) obtained in this way will likewise be called induced.

Under the projection of an irreducible space \(\Pi'\) of tensor fields on \(S_n\) onto \(K^*\), the preimage of the zero tensor field on \(K^*\) on \(S^*\) either fills the whole space \(\Pi'\), or coincides with the zero tensor field on \(S_n\). In the latter case the mapping under projection will be one-to-one, and the resulting space \(\Pi^*\) of tensor fields on \(K^*\) will be irreducible.

The set of irreducible spaces \(\Pi^*\) of induced tensor fields on \(K^*\) is complete in the sense that every continuous tensor field on \(K^*\) admits a uniform approximation by finite sums of the given induced tensor fields.

Indeed, every continuous tensor field \(P^*\) on \(K^*\) can be represented as the projection of a tensor field \(P'\), continuous on \(S_n\). If a sequence of tensor fields on \(S_n\) converges uniformly to the tensor field \(P'\), then the sequence of projections on \(K^*\) of these tensor fields also converges uniformly to the projection \(P^*\) of the tensor field \(P'\). Hence our assertion follows from Theorem 1 \((^2)\).

With this we finish the consideration of the general case and pass to Grassmann manifolds.

Thus,
\[ K = O(m)/O(m-k)O(k), \]
where \(O(m)\) is the proper orthogonal group.

As is known, the subgroup \(O(m-k)O(k)\) is singled out from the group \(O(m)\) by the condition of invariance of a simple \(k\)-vector (or of an \((m-k)\)-vector orthogonal to it).

For \(E_{n+1}^{1}\), where
\[ n+1=\frac{m!}{(m-k)!k!}, \]
we take the Euclidean space of \(k\)-vectors \(z^{i_1,\ldots,i_k}\) with metric
\[ (z_i,z_j)=g_{i_1,j_1}g_{i_2,j_2}\cdots g_{i_k,j_k} z^{i_1,i_2,\ldots,i_k}z^{j_1,j_2,\ldots,j_k}. \]
The space \(K^*\) will be the intersection of the manifold of simple \(k\)-vectors with the unit sphere \(S_n\).

From a tensor \(T\) of valence \(ks\) of the space \(E_m\) we can obtain a tensor \(\hat T\) in the space \(E_{n+1}\) in the following way: the indices of the tensor \(T\) are divided into \(s\) groups of \(k\) indices in each, and alternation is carried out within each group; each such group is taken as an index in the space of \(k\)-vectors. We thus obtain in \(E_{n+1}\) a tensor \(\hat T\) of valence \(s\) (provided only that \(\hat T\ne0\)). Such a tensor in \(E_{n+1}\) will be called induced. It can be shown, relying on tensors having only one coordinate different from zero, that every tensor in \(E_{n+1}\) is representable in the form of a sum of induced tensors.

If, for tensors \(T\) in \(E_m\) belonging to an irreducible space \(\Pi\), the indices are uniformly divided into groups, then the corresponding space \(\hat\Pi\) of tensors \(\hat T\) in \(E_{n+1}\) will be irreducible with respect to the group \(O(m)\), acting in the obvious way in \(E_{n+1}\). The correspondence between tensors of the spaces \(\Pi\) and \(\hat\Pi\) in each case will either be one-to-one, or the space \(\hat\Pi\) will be zero.

Proposition 1. If \(m\) and \(k\) are even, then on \(K\) there acts not \(O(m)\), but its factor by the symmetry about the origin, and in the space of tensor fields \(T^*\) on \(K^*\) there occur precisely those irreducible representations of the group \(O(m)\) which are equivalent to representations by means of irreducible tensors \(T\) in \(E_m\) with an even number of indices. If, however, at least one of the numbers \(m\) and \(k\) is odd, then every finite-dimensional irreducible representation of the group has an equivalent one among the irreducible representations in the space of tensor fields on
\[ K=O(m)/O(m-k)O(k). \]

The space of all tensors in \(E_m\) decomposes into subspaces irreducible with respect to the group \(O(m)\), the tensors of each of which are representable in the form of a product of some number \(s\) of metric tensors by a traceless tensor with a Young scheme \((^3)\).

For each such space the Young scheme and the number \(s\) of metric tensors are fixed. The set of such spaces \(\Pi\) of tensors \(T\) in \(E_m\) we shall call \(M\). It is complete in the sense that every tensor in \(E_m\) is representable as a finite sum of tensors, each of which belongs to some irreducible space—an element of \(M\). If, for each element of \(M\), the indices of the tensors belonging to it are divided into groups of \(k\) indices in each (if this is possible) in all possible ways and, after alternating over these groups, one obtains spaces of tensors in \(E_{n+1}\) irreducible with respect to the group \(\varphi(O(m))\), then the resulting set-

into \(M'\) of these spaces is complete in the preceding sense. In contrast to (3), we shall take the Young diagrams in skew-symmetric form.

It is of interest to note certain restrictions on the arrangement of groups of indices with respect to \(k\), in spite of which the obtained set \(M'\) preserves completeness.

Proposition 2. In order to preserve the completeness of the set \(M'\), it is sufficient to take only those partitions into groups for which no group contains two indices from one row of the Young diagram or two indices of one metric tensor.

From the set \(M'\) we can, by the method described above, obtain a set \(M^*\) of spaces of tensor fields on \(K^*\), irreducible with respect to the group \(\varphi(O(m))\), complete in the sense that any continuous tensor field on \(K^*\) admits a uniform approximation by finite sums of continuous tensor fields, each of which belongs to an irreducible space—an element of \(M^*\).

Moreover, it is of interest to consider also restrictions which, perhaps, destroy the completeness of \(M'\), but preserve the completeness of \(M^*\).

Theorem. Without violating the completeness of the set \(M^*\), among all possible ways of partitioning the indices of an element of \(\widehat M\) into groups one may take only those such that: 1) they satisfy Proposition 2; 2) each group, when projected (as an index of a tensor in \(E_{n+1}\)) subsequently onto the normal to \(S_n\), has no indices in rows of the Young diagram with number greater than \(k\); 3) each group of indices which, under projection onto \(S_n\), is projected onto the tangent plane has no more than one index from rows of the Young diagram with number greater than \(k\).

It follows from this that the number of indices in rows with number greater than \(k\) must not exceed \(\frac{1}{k}\) of the total number of indices of the tensor.

The system of restrictions obtained is, unfortunately, not yet complete. For example, for \(K = O(m)/O(m-2)O(2)\) the tensor with Young diagram

\[ \begin{matrix} 1 & 1\\ 3 & 4\\ 5 & 6 \end{matrix} \]

with groups of indices \(1,4;\ 3,6;\ 2,5\) satisfies all the conditions, but already under the mapping into \(M'\) a zero element is obtained.

Received
11 V 1963

REFERENCES

1. P. K. Rashevskii, Matem. sborn., 50 (92), 2, 171 (1960).
2. N. I. Glazunov, DAN, 148, No. 2, 264 (1963).
3. H. Weyl, Classical Groups, Moscow, 1947.