Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 2, 303-315

The Trace Decomposition Problem

Demeter Krupka^{1}

Department of Mathematics, Silesian University Opava Bezrucovo nam. 13, 746 01 Opava, Czech Republic E-mail address: Demeter.Krupka@fpf.slu.cz

Abstract. The problem of decomposition of mixed tensor spaces by the trace operation is considered. It is shown that a tensor A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) can always be expressed as the sum of a traceless term and a linear combination of the Kronecker’s δ-tensor, with traceless coefficients. The uniqueness of this decomposition is discussed. Particular cases of the trace decomposition of tensor spaces of type (1, 2), (1, 3), and (2, 2) including the cases of the torsion and curvature tensors, are considered, and explicit decomposition formulas are given. This analysis enables us to derive the Weyl projective, and Weyl conformal curvature tensors. Key words: Trace, traceless tensor, tensor space decomposition, torsion tensor, curvature tensor. MSC 1991: 15A72, 53A55

1. Introduction

Let E be a real, n-dimensional vector space, E_{q}^{p} the vector space of tensors of type (p, q) on E. We always use a fixed basis of E, and the induced basis of E_{q}^{p}. Thus, if A ∈ E_{q}^{p} is a

tensor, we write for simplicity A = (A_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}). We say that a tensor A ∈ E_{q}^{p} is traceless, if its traces are all zero. We formulate the trace decomposition problem as the problem of existence and uniqueness of a decomposition of a tensor in which one term is traceless, and the remaining terms are a linear combination of the Kronecker’s δ-tensor.

The problem of decomposition of tensors by the trace operation was studied by H. Weyl as a part of representation theory of the orthogonal group [6]. This paper is partially motivated by Weyl’s theory.

However, main reason for our study consists in investigation of the structure of modules of differential forms on jet prolongations of fibered manifolds which appear in global variational theory. The decompositions apply to components of forms, and result in criteria of contactness and strong contactness (see [1], [4]). Within this context, the trace

1 Research supported by grant No. 201/93/2245 from the Czech Grant Agency. decomposition problem appears as a problem of solving some systems of linear algebraic equations, with no reference to the group representation theory.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

In this paper we adopt this point of view. It is easily seen, however, that the resulting trace decompositions of tensors are automatically invariant with respect to the tensor representations of the general linear group. The corresponding invariant subspaces are not, in general, indecomposable. In fact, complete decompositions of a tensor space can be obtained by applying the Young-Kronecker theory to both contravariant, and covariant indices of every summand in its trace decomposition. It is not our aim in this paper to combine the trace decomposition theory and the Young-Kronecker theory (see [5]).

In Sec. 2 we prove two basic theorems providing a complete solution of the trace decomposition problem in its algebraic setting. The methods of the proof do not give us a general trace decomposition formula. In Sec. 3 we consider some particular cases which are important in differential geometry. We give explicit decomposition formulas for tensor spaces of types (1, 1), (1, 2), (1, 3), and (2, 2). As one would expect, the formulas describing the trace decomposition strongly depend on the dimension of the underlying vector space. In Sec. 4 we apply these formulas to the torsion and curvature tensors on a manifold endowed with a linear connection or a metric. Extensive literature on this subject from different perspectives can be found. For example, in [2], the case of a torsionfree connection is considered on a vector space endowed with a positive definite metric, and a complete decomposition of the underlying tensor space, invariant with respect to the group SO(n), is obtained. Our approach consists in a direct analysis of linear algebraic equations, defining the trace decompositions. We show that these decompositions lead to the Weyl projective and conformal curvature tensors well-known in differential geometry of linear connections and metric fields. These tensors are obtained as curvatures which are invariant under projective and conformal transformations respectively (see [7]). In this paper we show that the trace decomposition theory defines these tensors uniquely as traceless components in the trace decompositions of the corresponding tensor spaces. For complete proofs of the trace decomposition formulas for tensors of types (1, 2) and (1, 3) we refer to [3].

2. The trace decomposition

Let p, q, and n be positive integers such that p ≤ q, and let E_{q}^{p} denote the vector space

of systems of real numbers A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}), where 1 ≤ i_{1}, i_{2}, …, i_{p}, k_{1}, k_{2}, …, k_{q} ≤ n. A system A ∈ E_{q}^{p} is said to be of type (p, q).

A system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) ∈ E_{q}^{p} is called traceless, if its traces are all zero. A is called

δ^{1}-generated if there exist systems

(1)  \begin{aligned}B_{(s)}^{(r)} = & (B_{(s)}^{(r)}_{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}) ∈ E_{q−1}^{p−1}, 1 ≤ r ≤ p, 1 ≤ s ≤ q, \\ & 1 ≤ i_{1}, i_{2}, …, i_{p−1}, k_{1}, k_{2}, …, k_{q−1} ≤ n \end{aligned}

such that every element A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} can be expressed in the form

A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} = δ_{k_{1}}^{i_{1}}B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + δ_{k_{2}}^{i_{1}}B_{(2)}^{(1)} _{k_{1}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{1}}B_{(q)}^{(1)} _{k_{1}k_{2}...k_{q−1}}^{i_{2}i_{3}...i_{p}}+
δ_{k_{1}}^{i_{2}}B_{(1)}^{(2)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + δ_{k_{2}}^{i_{2}}B_{(2)}^{(2)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{2}}B_{(q)}^{(2)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{3}...i_{p}}+
…
δ_{k_{1}}^{i_{p}}B_{(1)}^{(p)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + δ_{k_{2}}^{i_{p}}B_{(2)}^{(p)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + … + δ_{k_{q}}^{i_{p}}B_{(q)}^{(p)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}.

The systems (1) are then called the δ^{1}-components of the the δ^{1}-generated system A. It is easily seen that the components as introduced above need not be defined uniquely. A δ^{1}-generated system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) is called δ^{2}-generated if it has δ^{1}-components that

are δ^{1}-generated (in E_{q−1}^{p−1}). Proceeding on induction, we define a δ^{r}-generated system, where 1 ≤ r ≤ p, as a δ^{r−1}-generated system whose δ^{1}-components are all δ^{r−1}-generated. To complete the definition, we call any system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) to be δ^{0}-generated. The

component of a δ^{0}-generated system is unique, and coincides with this system. Clearly, a δ^{r}-generated system is always δ^{r−1}-generated. It is now evident how to introduce the δ^{r}-components of a δ^{r}-generated system.

A δ^{r}-generated system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}), where 0 ≤ r ≤ p − 1, whose δ^{r}-components

are traceless (in E_{q−r}^{p−r}) is said to be relatively traceless. A δ^{0}-generated, relatively traceless system is traceless.

Let A ∈ E_{q}^{p}. The trace decomposition problem can now be formulated as the problem of existence and uniqueness of δ^{r}-generated, relatively traceless systems A^{(r)} ∈ E_{q}^{p}, where 0 ≤ r ≤ p−1, and a δ^{p}-generated system A^{(p)} ∈ E_{q}^{p} such that A = A^{(0)}+A^{(1)}+A^{(2)}+…+A^{(p)}. To solve this problem, we begin by proving the following result.

Theorem 1. Let p, q, n be positive integers, p ≤ q, and let A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) ∈ E_{q}^{p}.

There exist a system B = (B_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) and systems B_{(s)}^{(r)} = (B_{(s)}^{(r)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}), where 1 ≤ r ≤ p, 1 ≤ s ≤ q, such that

(2) \begin{aligned}A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} = & B_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} + δ_{k_{1}}^{i_{1}}B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + δ_{k_{2}}^{i_{1}}B_{(2)}^{(1)} _{k_{1}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{1}}B_{(q)}^{(1)} _{k_{1}k_{2}...k_{q−1}}^{i_{2}i_{3}...i_{p}}+ \\ & δ_{k_{1}}^{i_{2}}B_{(1)}^{(2)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + δ_{k_{2}}^{i_{2}}B_{(2)}^{(2)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{2}}B_{(q)}^{(2)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{3}...i_{p}}+ \\ & … \\ & δ_{k_{1}}^{i_{p}}B_{(1)}^{(p)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + δ_{k_{2}}^{i_{p}}B_{(2)}^{(p)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + … + δ_{k_{q}}^{i_{p}}B_{(q)}^{(p)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}} \end{aligned}

and

(3)  \begin{aligned}B_{sk_{2}...k_{q}}^{si_{2}...i_{p}} = 0, & B_{k_{1}sk_{3}...k_{q}}^{si_{2}...i_{p}} = 0, …, B_{k_{1}k_{2}...k_{q−1}s}^{si_{2}...i_{p}} = 0, \\ B_{sk_{2}...k_{q}}^{i_{1}si_{3}...i_{p}} = 0, & B_{k_{1}sk_{3}...k_{q}}^{i_{1}si_{3}...i_{p}} = 0, …, B_{k_{1}k_{2}...k_{q−1}s}^{i_{1}si_{3}...i_{p}} = 0, \\ & … \\ B_{sk_{2}...k_{q}}^{i_{1}i_{2}...i_{p−1}s} = 0, & B_{k_{1}sk_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}s} = 0, …, B_{k_{1}k_{2}...k_{q−1}s}^{i_{1}i_{2}...i_{p−1}s} = 0. \end{aligned}

The system B is unique. If p + q ≤ n + 1, the systems B_{(s)}^{(r)} are also unique.

Proof. Let g = (g_{ij}) be any regular, positive definite matrix of dimension n × n defining a scalar product on R^{n}. g defines a scalar product on E_{q}^{p} by the formula

(4)  (A|B) = g_{i_{1}j_{1}}g_{i_{2}j_{2}} … g_{i_{p}j_{p}}g^{k_{1}l_{1}}g^{k_{2}l_{2}} … g^{k_{q}l_{q}}A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}B_{l_{1}l_{2}...l_{q}}^{j_{1}j_{2}...j_{p}}.

Consider the subspace of E_{q}^{p} generated by the systems X = (X_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}) of the form

X_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} = δ_{k_{1}}^{i_{1}}X_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + δ_{k_{2}}^{i_{1}}X_{(2)}^{(1)} _{k_{1}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{1}}X_{(q)}^{(1)} _{k_{1}k_{2}...k_{q−1}}^{i_{2}i_{3}...i_{p}}+
δ_{k_{1}}^{i_{2}}X_{(1)}^{(2)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + δ_{k_{2}}^{i_{2}}X_{(2)}^{(2)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{2}}X_{(q)}^{(2)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{3}...i_{p}}+
…
δ_{k_{1}}^{i_{p}}X_{(1)}^{(p)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + δ_{k_{2}}^{i_{p}}X_{(2)}^{(p)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + … + δ_{k_{q}}^{i_{p}}X_{(q)}^{(p)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}.

The orthogonal complement of this subspace is formed by all systems B = (B_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}) such that (B|X) = 0 for all X. Expressing the left side explicitly, we easily see that the orthogonal complement consists of the traceless systems B = (B_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}) ∈ E_{q}^{p}.

If now A = (A_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}) is an arbitrary element of E_{q}^{p}, we can express it as the sum of its projections onto these complementary orthogonal subspaces. This proves the existence of the desired decomposition. The uniqueness of the system B = (B_{j_{1}j_{2}...j_{q}}^{i_{1}i_{2}...i_{p}}) is obvious.

Now we prove that if p + q ≤ n + 1, then the systems B_{(s)}^{(r)} = (B_{(s)}^{(r)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}) are unique. We wish to show that under this assumption, the system (2), (3) has a unique solution. It is sufficient to show that the homogeneous system

(5)  \begin{matrix}δ_{k_{1}}^{i_{1}}B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + δ_{k_{2}}^{i_{1}}B_{(2)}^{(1)} _{k_{1}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{1}}B_{(q)}^{(1)} _{k_{1}k_{2}...k_{q−1}}^{i_{2}i_{3}...i_{p}}+ \\ δ_{k_{1}}^{i_{2}}B_{(1)}^{(2)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + δ_{k_{2}}^{i_{2}}B_{(2)}^{(2)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}} + … + δ_{k_{q}}^{i_{2}}B_{(q)}^{(2)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{3}...i_{p}}+ \\ … \\ δ_{k_{1}}^{i_{p}}B_{(1)}^{(p)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + δ_{k_{2}}^{i_{p}}B_{(2)}^{(p)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}} + … + δ_{k_{q}}^{i_{p}}B_{(q)}^{(p)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}} = 0 \end{matrix}

has the trivial solution only. Choose a basis ξ_{1}, ξ_{2}, …, ξ_{n} of the vector space R^{n}, and write in components (with respect to the canonical basis) ξ_{j} = (ξ_{j}^{i}). Analogously, denote by ω^{1}, ω^{2}, …, ω^{n} elements of the dual basis of the dual vector space R^{n∗}, and write ω^{i} = (ω_{k}^{i}). By definition,

(6)  ξ_{i}^{j}ω_{k}^{i} = δ_{k}^{j}, 1 ≤ j, k ≤ n.

Choose an index j such that 1 ≤ j ≤ n. Since p + q ≤ n + 1, one can always choose indices r_{1}, r_{2}, …, r_{p−1} and s_{1}, s_{2}, …, s_{q−1}, all mutually different, such that j ≠ r_{1}, r_{2}, …, r_{p−1}, s_{1}, s_{2}, …, s_{q−1}. Let for example j be equal to 1. Then r_{1} = 2, r_{2} = 3, …, r_{p−1} = p and s_{1} = p+1, s_{2} = p+2, …, s_{q−1} = p+q −1, where by hypothesis, p+q −1 ≤ n. Multiplying (5) by

w_{i_{1}i_{2}...i_{p}}^{k_{1}k_{2}...k_{q}} = ξ_{1}^{k_{1}}ξ_{p+1}^{k_{p+1}}ξ_{p+2}^{k_{p+2}} … ξ_{p+q−1}^{k_{p+q−1}}ω_{i_{1}}^{1}ω_{i_{2}}^{2} … ω_{i_{p}}^{p}

we obtain

B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}}w_{si_{2}i_{3}...i_{p}}^{sk_{2}...k_{q}} + B_{(2)}^{(1)} _{k_{1}k_{3}...k_{q}}^{i_{2}...i_{p}}w_{si_{2}i_{3}...i_{p}}^{k_{1}s...k_{q}} + … + B_{(q)}^{(1)} _{k_{1}k_{2}...k_{q−1}}^{i_{2}...i_{p}}w_{si_{2}i_{3}...i_{p}}^{k_{1}k_{2}...k_{q−1}s}+
B_{(1)}^{(2)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}}w_{i_{1}si_{3}...i_{p}}^{sk_{2}...k_{q}} + B_{(2)}^{(2)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{3}...i_{p}}w_{i_{1}si_{3}...i_{p}}^{k_{1}s...k_{q}} + … + B_{(q)}^{(2)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{3}...i_{p}}w_{i_{1}si_{3}...i_{p}}^{k_{1}k_{2}...k_{q−1}s}+
…

B_{(1)}^{(p)} _{k_{2}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}}w_{i_{1}i_{2}...i_{p−1}s}^{sk_{2}...k_{q}} + B_{(2)}^{(p)} _{k_{1}k_{3}...k_{q}}^{i_{1}i_{2}...i_{p−1}}w_{i_{1}i_{2}...i_{p−1}s}^{k_{1}s...k_{q}} + … + B_{(q)}^{(p)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}w_{i_{1}i_{2}...i_{p−1}s}^{k_{1}k_{2}...k_{q−1}s} = 0.

Using (6) this equation reduces to

(7)  B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}}ξ_{p+1}^{k_{2}}ξ_{p+2}^{k_{3}} … ξ_{p+q−1}^{k_{q}}ω_{i_{2}}^{2} … ω_{i_{p}}^{p} = 0.

Since (7) holds for all ξ_{p+1}^{k_{2}}, ξ_{p+2}^{k_{3}}, …, ξ_{p+q−1}^{k_{q}}, ω_{i_{2}}^{2}, ω_{i_{3}}^{3}, …, ω_{i_{p}}^{p}, we have

B_{(1)}^{(1)} _{k_{2}k_{3}...k_{q}}^{i_{2}i_{3}...i_{p}} = 0.

Proceeding in the same way we obtain

B_{(s)}^{(r)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}} = 0, 1 ≤ r ≤ p, 1 ≤ s ≤ q,

which means that the system (5) has the trivial solution only.

Formulas (2) and (3) of Theorem 1 state the existence and uniqueness of the first trace decomposition of a system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) ∈ E_{q}^{p}.

Remark. Since the traceless component B ∈ E_{q}^{p} of A ∈ E_{q}^{p} is unique, its complementary component is also unique. However, this fact does not imply, in general, the uniqueness of the systems B_{(s)}^{(r)} ∈ E_{q−1}^{p−1}. Example p = 1, q = 3, n = 2 shows that if the inequality

p + q ≤ n + 1 is violated, then the systems B_{(s)}^{(r)} might not be unique (see Sec. 3).

Now we are in a position to formulate the following result.

Theorem 2. Let p, q, and n be positive integers such that p ≤ q, and let A ∈ E_{q}^{p}. There exist unique δ^{r}-generated, relatively traceless systems A^{(r)} ∈ E_{q}^{p}, where 0 ≤ r ≤ p − 1, and a δ^{(p)}-generated system A^{(p)} ∈ E_{q}^{p}, such that

(8)  A = A^{(0)} + A^{(1)} + A^{(2)} + … + A^{(p)}.

Proof. Consider a system A = (A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}}) ∈ E_{q}^{p}. By Theorem 1, we express A_{k_{1}k_{2}...k_{q}}^{i_{1}i_{2}...i_{p}} in

the form (2). Then we can apply the decomposition (2) to each of the systems B_{(s)}^{(r)} =

(B_{(s)}^{(r)} _{k_{1}k_{2}...k_{q−1}}^{i_{1}i_{2}...i_{p−1}}). Repeating this procedure p times we obtain the desired decomposition (8). This proves existence.

To prove uniqueness, it is sufficient to show that for every pair of indices m, t such that 0 ≤ m < t ≤ p, the components A^{(m)}, A^{(t)} of A are orthogonal (with respect to the scalar product (4) on E_{q}^{p}). If m = 0, the orthogonality has already been shown in the proof of Theorem 1. Let m, t ≥ 1. Then the system A^{(m)} is a sum of terms of the form

δ_{k_{s_{1}}}^{i_{r_{1}}} δ_{k_{s_{2}}}^{i_{r_{2}}} … δ_{k_{sm}}^{i_{rm}} A_{(s_{1}s_{2}...s_{m})}^{(r_{1}r_{2}...r_{m})} _{k_{s_{m+1}} k_{s_{m+2}} ...k_{sq}}^{i_{r_{m+1}} i_{r_{m+2}} ...i_{rp}} ,

and A^{(t)} is a sum of terms of the form

δ_{k_{s_{1}}}^{i_{r_{1}}} δ_{k_{s_{2}}}^{i_{r_{2}}} … δ_{k_{s_{t}}}^{i_{r_{t}}} A_{(s_{1}s_{2}...s_{t})}^{(r_{1}r_{2}...r_{t})} _{k_{s_{t+1}} k_{s_{t+2}} ...k_{sq}}^{i_{r_{t+1}} i_{r_{t+2}} ...i_{rp}} .

Since A^{(m)} is traceless, we get after some calculation

(9)  (A^{(m)}|A^{(t)}) = 0

for all m and t proving the uniqueness.

Formula (8) of Theorem 2 states the second trace decomposition of a system A ∈ E_{q}^{p}.

3. The trace decomposition of systems of type (1,1), (1,2), (1,3), and (2,2)

In this section we give trace decomposition formulas of a system A ∈ E_{q}^{p} in the following four cases: 3.1. p = 1, q = 1, 3.2. p = 1, q = 2, 3.3. p = 1, q = 3, and 3.4. p = 2, q = 2. We discuss the uniqueness of the decomposition.

The proofs are based on a direct analysis of the corresponding systems of linear equations, and are tedious and long. Therefore, we omit the proofs. In cases 3.1, 3.2, and 3.3, we proceed in three steps: 1) we derive the first contracted system by forming all possible contractions of the given system of linear equations, 2) we solve the first contracted system, and 3) we substitute the result into the initial system (see [3]). In the case 3.4, we have one more step: 1) we derive the first contracted system, 2) contracting the first contracted system we derive the second contracted system, 3) we solve the second contracted system and substitute the result into the first one, 4) we solve the first contracted system and substitute the result into the initial system.

Note that if n = 1, the trace decomposition is trivial: any trace of a system consisting of one element coincides with this element. Consequently, traceless systems are all zero, and the trace decomposition is not unique unless p = q = 1. In what follows we always assume n ≥ 2.

3.1. In the case p = q = 1 we have the following evident decomposition formulas presented here for completeness.

Theorem 3. Let p = q = 1, n ≥ 2, and let A = (A_{k}^{i}) ∈ E_{1}^{1}. Then there exist a unique traceless system B = (B_{k}^{i}) ∈ E_{1}^{1} and a unique number C ∈ R such that

A_{k}^{i} = B_{k}^{i} + δ_{k}^{i}C.

These systems are defined by

C = \frac{1}{n}A_{s}^{s}, B_{k}^{i} = A_{k}^{i} − \frac{1}{n}δ_{k}^{i}A_{s}^{s}.

3.2. If p = 1, q = 2, then the trace decomposition problem has a unique solution.

Theorem 4. Let p = 1, q = 2, n ≥ 2, and let A = (A_{kl}^{i}) ∈ E_{2}^{1}. Then there exist a unique traceless system B = (B_{kl}^{i}) ∈ E_{2}^{1} and unique systems C = (C_{k}), D = (D_{k}) ∈ E_{1}^{0} such that

A_{kl}^{i} = B_{kl}^{i} + δ_{k}^{i}C_{l} + δ_{l}^{i}D_{k}.

These systems are defined by

C_{l} = \frac{1}{n^{2} − 1}(nA_{tl}^{t} − A_{lt}^{t}), D_{k} = \frac{1}{n^{2} − 1}(−A_{tk}^{t} + nA_{kt}^{t}),

B_{kl}^{i} = A_{kl}^{i} − \frac{1}{n^{2} − 1}(δ_{k}^{i}(nA_{tl}^{t} − A_{lt}^{t}) + δ_{l}^{i}(−A_{tk}^{t} + nA_{kt}^{t})).

3.3. Let p = 1, q = 3. In this case the trace decomposition problem has always a solution; if n ≥ 3, the solution is unique.

Theorem 5. Let p = 1, q = 3, and let A = (A_{klm}^{i}) ∈ E_{3}^{1}.

(a) Assume that n = 2. Then there exist a unique traceless system B = (B_{klm}^{i}) ∈ E_{3}^{1} and systems C = (C_{lm}), D = (D_{km}), E = (E_{kl}) ∈ E_{2}^{0} such that

A_{klm}^{i} = B_{klm}^{i} + δ_{k}^{i}C_{lm} + δ_{l}^{i}D_{km} + δ_{m}^{i}E_{kl}.

These systems are defined by

C_{21} = −\frac{1}{3}(A_{t12}^{t} − λ) + \frac{2}{3}A_{1t2}^{t} − A_{12t}^{t} + \frac{4}{3}A_{t21}^{t} − \frac{2}{3}(A_{2t1}^{t} − λ),

C_{22} = \frac{1}{4}(3A_{t22}^{t} − A_{2t2}^{t} − A_{22t}^{t}),

D_{11} = \frac{1}{4}(−A_{t11}^{t} + 3A_{1t1}^{t} − A_{11t}^{t}),

D_{12} = −\frac{2}{3}(A_{t12}^{t} − λ) + \frac{4}{3}A_{1t2}^{t} − A_{12t}^{t} + \frac{2}{3}A_{t21}^{t} − \frac{1}{3}(A_{2t1}^{t} − λ),

D_{21} = \frac{1}{6}(A_{t12}^{t} − λ) − \frac{1}{3}A_{1t2}^{t} + \frac{1}{2}A_{12t}^{t} − \frac{2}{3}A_{t21}^{t} + \frac{5}{6}(A_{2t1}^{t} − λ),

D_{22} = \frac{1}{4}(−A_{t22}^{t} + 3A_{2t2}^{t} − A_{22t}^{t}),

E_{11} = \frac{1}{4}(−A_{t11}^{t} − A_{1t1}^{t} + 3A_{11t}^{t}),

E_{12} = \frac{1}{2}(A_{t12}^{t} − λ) − A_{1t2}^{t} + \frac{3}{2}A_{12t}^{t} − A_{t21}^{t} + \frac{1}{2}(A_{2t1}^{t} − λ),

E_{21} = λ,

E_{22} = \frac{1}{4}(−A_{t22}^{t} − A_{2t2}^{t} + 3A_{22t}^{t}),

where λ ∈ R is a parameter.

(b) Assume that n ≥ 3. Then there exist a unique traceless system B = (B_{klm}^{i}) ∈ E_{3}^{1} and unique systems C = (C_{lm}), D = (D_{km}), E = (E_{kl}) ∈ E_{2}^{0} such that

A_{klm}^{i} = B_{klm}^{i} + δ_{k}^{i}C_{lm} + δ_{l}^{i}D_{km} + δ_{m}^{i}E_{kl}.

These systems are defined by

C_{kl} = \frac{n(n^{2}− 3)A_{tkl}^{t} − (n^{2}− 2)A_{ktl}^{t} + nA_{klt}^{t} − 2A_{tlk}^{t} + nA_{ltk}^{t} − (n^{2}− 2)A_{lkt}^{t}}{(n^{2}− 1)(n^{2}− 4)} ,

D_{kl} = \frac{−(n^{2}− 2)A_{tkl}^{t} + n(n^{2}− 3)A_{ktl}^{t} − (n^{2}− 2)A_{klt}^{t} + nA_{tlk}^{t} − 2A_{ltk}^{t} + nA_{lkt}^{t}}{(n^{2}− 1)(n^{2}− 4)} ,

E_{kl} = \frac{nA_{tkl}^{t} − (n^{2}− 2)A_{ktl}^{t} + n(n^{2}− 3)A_{klt}^{t} − (n^{2}− 2)A_{tlk}^{t} + nA_{ltk}^{t} − 2A_{lkt}^{t}}{(n^{2}− 1)(n^{2}− 4)} ,

and

B_{klm}^{i} = A_{klm}^{i} − δ_{k}^{i}C_{lm} − δ_{l}^{i}D_{km} − δ_{m}^{i}E_{kl}.

3.4. Let p = 2, q = 2. In this case the trace decomposition problem has always a solution; if n ≥ 3, the solution is unique.

Theorem 6. Let p = 2, q = 2, and let A = (A_{kl}^{ij}) ∈ E_{2}^{2}.

(a) Assume that n = 2. Then there exist a unique traceless system B = (B_{kl}^{ij}) ∈ E_{2}^{2}, traceless systems C = (C_{l}^{j}), D = (D_{k}^{j}), E = (E_{l}^{i}), F = (F_{k}^{i}) ∈ E_{1}^{1}, and unique numbers G, H ∈ R such that

A_{kl}^{ij} = B_{kl}^{ij} + δ_{k}^{i}C_{l}^{j} + δ_{l}^{i}D_{k}^{j} + δ_{k}^{j}E_{l}^{i} + δ_{l}^{j}F_{k}^{i} + δ_{k}^{i}δ_{l}^{j}G + δ_{l}^{i}δ_{k}^{j}H

for all i, j, k, l = 1, 2. These systems are defined by

G = \frac{1}{6} (2A_{st}^{st} − A_{ts}^{st}) , H = \frac{1}{6} (−A_{st}^{st} + 2A_{ts}^{st}) ,

C_{l}^{i} = A_{sl}^{si} − \frac{1}{2}A_{ls}^{si} − \frac{1}{2}A_{sl}^{is} − \frac{1}{2}δ_{l}^{i}(A_{st}^{st} − A_{ts}^{st}) + F_{l}^{i},

D_{k}^{j} = −\frac{1}{2}A_{sk}^{sj} + \frac{3}{4}A_{ks}^{sj} + \frac{1}{4}A_{sk}^{js} + \frac{1}{4}δ_{k}^{j}(A_{st}^{st} + 2A_{ts}^{st}) − F_{k}^{j},

E_{l}^{i} = −\frac{1}{2}A_{sl}^{si} + \frac{1}{4}A_{ls}^{si} + \frac{3}{4}A_{sl}^{is} + \frac{3}{4}δ_{l}^{i}(A_{st}^{st} − 2A_{ts}^{st}) − F_{l}^{i},

and

B_{kl}^{ij} = A_{kl}^{ij} − δ_{k}^{i} (A_{sl}^{sj} − \frac{1}{2}A_{ls}^{sj} − \frac{1}{2}A_{sl}^{js}) − δ_{l}^{i} (−\frac{1}{2}A_{sk}^{sj} + \frac{3}{4}A_{ks}^{sj} + \frac{1}{4}A_{sk}^{js})

−δ_{k}^{j} (−\frac{1}{2}A_{sl}^{si} + \frac{1}{4}A_{ls}^{si} + \frac{3}{4}A_{sl}^{is}) + \frac{1}{6}δ_{k}^{i}δ_{l}^{j}A_{st}^{st} − \frac{1}{3}δ_{l}^{i}δ_{k}^{j}A_{st}^{st} − \frac{1}{3}δ_{k}^{i}δ_{l}^{j}A_{ts}^{st} + \frac{2}{3}δ_{l}^{i}δ_{k}^{j}A_{ts}^{st},

where either F_{1}^{1}, F_{2}^{1}, F_{1}^{2} ∈ R or F_{2}^{2}, F_{2}^{1}, F_{1}^{2} ∈ R are parameters such that F_{1}^{1} + F_{2}^{2} = 0.

(b) Assume that n ≥ 3. Then there exist unique traceless systems B = (B_{kl}^{ij}) ∈ E_{2}^{2}, C = (C_{l}^{j}), D = (D_{k}^{j}), E = (E_{l}^{i}), F = (F_{k}^{i}) ∈ E_{1}^{1}, and unique numbers G, H ∈ R such that

A_{kl}^{ij} = B_{kl}^{ij} + δ_{k}^{i}C_{l}^{j} + δ_{l}^{i}D_{k}^{j} + δ_{k}^{j}E_{l}^{i} + δ_{l}^{j}F_{k}^{i} + δ_{k}^{i}δ_{l}^{j}G + δ_{l}^{i}δ_{k}^{j}H.

These systems are defined by

G = \frac{1}{n(n^{2} − 1)}(nA_{st}^{st} − A_{ts}^{st}), H = \frac{1}{n(n^{2} − 1)}(−A_{st}^{st} + nA_{ts}^{st}),

C_{l}^{j} = \frac{1}{n(n^{2} − 4)}((n^{2} − 2)A_{sl}^{sj} − nA_{ls}^{sj} − nA_{sl}^{js} + 2A_{ls}^{js} − nδ_{l}^{j}A_{st}^{st} + 2δ_{l}^{j}A_{ts}^{st}),

D_{l}^{j} = \frac{1}{n(n^{2} − 4)}(−nA_{sl}^{sj} + (n^{2} − 2)A_{ls}^{sj} + 2A_{sl}^{js} − nA_{ls}^{js} + 2δ_{l}^{j}A_{st}^{st} − nδ_{l}^{j}A_{ts}^{st}),

E_{l}^{j} = \frac{1}{n(n^{2} − 4)}(−nA_{sl}^{sj} + 2A_{ls}^{sj} + (n^{2} − 2)A_{sl}^{js} − nA_{ls}^{js} + 2δ_{l}^{j}A_{st}^{st} − nδ_{l}^{j}A_{ts}^{st}),

F_{l}^{j} = \frac{1}{n(n^{2} − 4)}(2A_{sl}^{sj} − nA_{ls}^{sj} − nA_{sl}^{js} + (n^{2} − 2)A_{ls}^{js} − nδ_{l}^{j}A_{st}^{st} + 2δ_{l}^{j}A_{ts}^{st}),

and

B_{kl}^{ij} = A_{kl}^{ij} − \frac{1}{n(n^{2} − 4)}δ_{k}^{i}((n^{2} − 2)A_{sl}^{sj} − nA_{ls}^{sj} − nA_{sl}^{js} + 2A_{ls}^{js})

−\frac{1}{n(n^{2} − 4)}δ_{l}^{i}(−nA_{sk}^{sj} + (n^{2} − 2)A_{ks}^{sj} + 2A_{sk}^{js} − nA_{ks}^{js})

−\frac{1}{n(n^{2} − 4)}δ_{k}^{j}(−nA_{sl}^{si} + 2A_{ls}^{si} + (n^{2} − 2)A_{sl}^{is} − nA_{ls}^{is})

−\frac{1}{n(n^{2} − 4)}δ_{l}^{j}(2A_{sk}^{si} − nA_{ks}^{si} − nA_{sk}^{is} + (n^{2} − 2)A_{ks}^{is})

−\frac{1}{(n^{2} − 4)(n^{2} − 1)}δ_{k}^{i}δ_{l}^{j}(−(n^{2} + 2)A_{st}^{st} + 3nA_{ts}^{st})

−\frac{1}{(n^{2} − 4)(n^{2} − 1)}δ_{l}^{i}δ_{k}^{j}(3nA_{st}^{st} − (n^{2} + 2)A_{ts}^{st}).

4. Applications: The trace decomposition of torsion and curvature tensors

In this section, X is a smooth n-dimensional manifold, where n ≥ 3. We apply the trace decomposition formulas of Theorems 4, 5, and 6 to tensors of type (1,2), (1,3), and (2,2) having the same symmetry properties as the torsion and curvature tensors of a linear connection on X.

4.1. The torsion tensor of a linear connection

Consider a torsion tensor T on X. T is a tensor field of type (1,2) written in a chart on X with coordinates (x^{1}, x^{2}, …, x^{n}) as

T = T_{jk}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j} ⊗ dx^{k},

where the components T_{jk}^{i} satisfy T_{jk}^{i} + T_{kj}^{i} = 0. The trace decomposition formula of T is given by Theorem 4. We get T_{kl}^{i} = B_{kl}^{i} + δ_{k}^{i}C_{l} + δ_{l}^{i}D_{k}, where

C_{l} = \frac{1}{n − 1}T_{tl}^{t}, D_{k} = −\frac{1}{n − 1}T_{tk}^{t}, B_{kl}^{i} = T_{kl}^{i} − \frac{1}{n − 1} (δ_{k}^{i}T_{tl}^{t} − δ_{l}^{i}T_{tk}^{t}) .

4.2. The curvature tensor of a linear connection

A curvature tensor R on X is a tensor field of type (1,3) written in a chart with coordinates (x^{1}, x^{2}, …, x^{n}) as

(10)  R = R_{jkl}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j} ⊗ dx^{k} ⊗ dx^{l},

where the components R_{jkl}^{i} satisfy R_{klm}^{i} + R_{kml}^{i} = 0. Recall that the Ricci tensor of R is defined as the (1,2)-trace of R, i.e., R_{kl} = R_{ksl}^{s} = −R_{kls}^{s}.

Using Theorem 5, the trace decomposition of R is obtained by a straightforward, but tedious calculation. We get R_{klm}^{i} = B_{klm}^{i} + δ_{k}^{i}C_{lm} + δ_{l}^{i}D_{km} + δ_{m}^{i}E_{kl}, where

C_{kl} = \frac{n − 1}{(n + 1)(n − 2)}R_{tkl}^{t} − \frac{1}{(n + 1)(n − 2)}(R_{kl} − R_{lk}),

D_{kl} = \frac{1}{(n^{2} − 1)(n − 2)} (−(n − 1)R_{tkl}^{t} + (n^{2} − n − 1)R_{kl} − R_{lk}) ,

E_{kl} = −D_{kl},

and

B_{klm}^{i} = R_{klm}^{i} − \frac{1}{(n + 1)(n − 2)}δ_{k}^{i} ((n − 1)R_{tlm}^{t} − R_{lm} + R_{ml})

−\frac{1}{(n^{2} − 1)(n − 2)}δ_{l}^{i} (−(n − 1)R_{tkm}^{t} + (n^{2} − n − 1)R_{km} − R_{mk})

−\frac{1}{(n^{2} − 1)(n − 2)}δ_{m}^{i} ((n − 1)R_{tkl}^{t} − (n^{2} − n − 1)R_{kl} + R_{lk}) .

4.3. The Weyl projective curvature tensor

Now consider a tensor field R (10) whose components satisfy R_{klm}^{i} + R_{kml}^{i} = 0, R_{klm}^{i} + R_{mkl}^{i} + R_{lmk}^{i} = 0. These symmetry conditions correspond to the curvature tensor of a torsionfree connection on X.

The (1,1)-trace of R is expressed by means of the Ricci tensor as R_{skl}^{s} = R_{kl} − R_{lk}. The trace decomposition of R has the form R_{klm}^{i} = B_{klm}^{i} + δ_{k}^{i}C_{lm} + δ_{l}^{i}D_{km} + δ_{m}^{i}E_{kl}, where

C_{kl} = \frac{1}{n + 1}(R_{kl} − R_{lk}), D_{kl} = \frac{1}{n^{2} − 1}(R_{lk} + nR_{kl}), E_{kl} = −\frac{1}{n^{2} − 1}(R_{lk} + nR_{kl}),

and

B_{klm}^{i} = R_{klm}^{i} − \frac{1}{n + 1}δ_{k}^{i}(R_{lm} − R_{ml}) − \frac{1}{n^{2} − 1}δ_{l}^{i}(R_{mk} + nR_{km}) + \frac{1}{n^{2} − 1}δ_{m}^{i}(R_{lk} + nR_{kl}).

This formula defines the well-known Weyl projective curvature tensor.

4.4. The Weyl projective curvature tensor of the Levi-Civita connection

Now let X be endowed with a regular, symmetric tensor field g of type (0,2) (the metric tensor); g may be a Riemann metric, but one may also consider indefinite metrics. In a chart on X with coordinates (x^{1}, x^{2}, …, x^{n}), g is expressed as g = g_{ij}dx^{i} ⊗ dx^{j}, where g_{ij} = g_{ji} and det(g_{ij}) ≠ 0. As usual, we denote by g^{ij} the components of the inverse matrix to the matrix (g_{ij}). Then g^{ij} = g^{ji}, det(g^{ij}) ≠ 0, and g_{ij}g^{jk} = g^{kj}g_{ji} = δ_{i}^{k}.

Consider a tensor R (10), and the tensor field of type (0,4)

(11)  R′ = R_{ijkl}dx^{i} ⊗ dx^{j} ⊗ dx^{k} ⊗ dx^{l},

where

(12)  R_{ijkl} = g_{im}R_{jkl}^{m}, R_{jkl}^{i} = g^{is}R_{sjkl}.

Assume that the following identities are satisfied:

(13) R_{ijkl} = −R_{ijlk}, R_{ijkl} = −R_{jikl}, R_{ijkl} = R_{klij}, R_{ijkl} + R_{iljk} + R_{iklj} = 0.

Because of these identities, the Ricci tensor is symmetric,

(14)  R_{jl} = R_{jsl}^{s} = g^{sk}R_{kjsl} = g^{sk}R_{sjkl} = g^{sk}R_{klsj} = R_{lj},

and

(15)  R_{tkl}^{t} = g^{st}R_{stkl} = 0.

We note that the identities (13) are satisfied by the curvature tensor of the Levi-Civita connection of g.

The trace decomposition of R has the form R_{klm}^{i} = B_{klm}^{i} + δ_{l}^{i}D_{km} + δ_{m}^{i}E_{kl}, where

D_{kl} = −E_{kl} = \frac{1}{n − 1}R_{kl}, B_{klm}^{i} = R_{klm}^{i} − \frac{1}{n − 1} (δ_{l}^{i}R_{km} − δ_{m}^{i}R_{kl}) .

The second formula defines the Weyl projective curvature tensor of the Levi-Civita connection.

4.5. The Weyl conformal curvature tensor

As in Sec. 4.4, we assume that X is endowed with a regular, symmetric tensor field g of type (0,2). Notation and formulas (11)–(15) remain valid. Recall that the scalar curvature of g is defined by r = g^{kl}g^{st}R_{sktl}.

Let R be a tensor (10) whose associated tensor R′ (11) satisfies (13). We wish to apply the trace decomposition formula of Theorem 6 to the tensor

(16)  R_{kl}^{ij} = g^{js}R_{skl}^{i} = g^{ir}g^{js}g_{rp}R_{skl}^{p} = g^{ir}g^{js}R_{rskl}.

We get R_{kl}^{ij} = B_{kl}^{ij} + δ_{k}^{i}C_{l}^{j} + δ_{l}^{i}D_{k}^{j} + δ_{k}^{j}E_{l}^{i} + δ_{l}^{j}F_{k}^{i} + δ_{k}^{i}δ_{l}^{j}G + δ_{l}^{i}δ_{k}^{j}H, where

(17)  \begin{matrix}G = \frac{1}{n(n − 1)}r, H = −\frac{1}{n(n − 1)}r, \\ C_{l}^{j} = F_{l}^{j} = −D_{l}^{j} = −E_{l}^{j} = \frac{1}{n(n − 2)}(ng^{jm}R_{ml} − δ_{l}^{j}r), \end{matrix}

and

(18)  \begin{matrix}B_{kl}^{ij} = R_{kl}^{ij} − & \frac{1}{n − 2}(δ_{k}^{i}g^{jm}R_{ml} − δ_{l}^{i}g^{jm}R_{mk} − δ_{k}^{j}g^{im}R_{ml} + δ_{l}^{j}g^{im}R_{mk}) \\ + & \frac{1}{(n − 1)(n − 2)}(δ_{k}^{i}δ_{l}^{j} − δ_{l}^{i}δ_{k}^{j})r. \end{matrix}

Formulas (17) and (18) represent the desired trace decomposition of the tensor (16).

If we use these formulas in the identity R_{pqkl} = g_{ip}g_{jq}R_{kl}^{ij} and define B_{pqkl} = g_{ip}g_{jq}B_{kl}^{ij}, we get the Weyl conformal curvature tensor

\begin{matrix}B_{pqkl} = R_{pqkl} − & \frac{1}{n − 2} (g_{kp}R_{ql} − g_{lp}R_{qk} − g_{kq}R_{pl} + g_{lq}R_{pk}) \\ + & \frac{1}{(n − 1)(n − 2)} (g_{kp}g_{kq} − g_{lp}g_{kq}) r. \end{matrix}

This formula shows us that the Weyl conformal curvature tensor is precisely the traceless component of the curvature tensor of a metric, considered as a tensor of type (2,2).

References

[1] Anderson, I. M.; Duchamp, T.: On the existence of global variational principles. Amer. Math. J. 102 (1980), 181–202.

[2] Bokan, N.: On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection. Rendiconti del Circ. Mat. di Palermo, Ser. II, 39 (1990), 331–380.

[3] Krupka, D.: The trace decomposition of tensors of type (1,2) and (1,3). Coll. Math. János Bolyai, Proc. Conf., July 1994, Debrecen, Hungary, J. Szenthe and L. Tamássy, Editors; Reidel, to appear.

[4] Krupka, D.: Topics in the calculus of variations: Variational sequences. Differential Geometry and Its Applications, Proc. Conf., Silesian Univ. Opava, Czech Republic, 1993, 473–495.

[5] Welsh, T. A.: Young Tableaux and Modules of Groups and Lie Algebras. Ph.D. thesis, University of Southampton, 1992.

[6] Weyl, H.: The Classical Groups. Princeton Univ. Press, Princeton, 1946.

[7] Weyl, H.: Zur Infinitesimalgeometrie, Einordnung der projektiven und der konformen Auffassung. Göttinger Nachrichten (1921), 99–112.

Received January 12, 1995; revised version June 10, 1995.