Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 217-231.

Natural Projectors in Tensor Spaces^{∗}

Demeter Krupka

Institute of Mathematics, Silesian University in Opava Bezrucovo nam. 13, 746 01 Opava, Czech Republic e-mail: Demeter.Krupka@math.slu.cz

Abstract. The aim of this paper is to introduce a method of invariant decompositions of the tensor space T_{s}^{r}R^{n} = R^{n} ⊗ R^{n} ⊗ ⋯ ⊗ R^{n} ⊗ R^{n∗} ⊗ R^{n∗} ⊗ ⋯ ⊗ R^{n∗} (r factors R^{n}, s factors the dual vector space R^{n∗}), endowed with the tensor representation of the general linear group GL_{n}(R). The method is elementary, and is based on the concept of a natural (GL_{n}(R)-equivariant) projector in T_{s}^{r}R^{n}. The case r = 0 corresponds with the Young-Kronecker decompositions of T_{s}^{0}R^{n} into its primitive components. If r, s ≠ 0, a new, unified invariant decomposition theory is obtained, including as a special case the decomposition theory of tensor spaces by the trace operation. As an example we find the complete list of natural projectors in T_{2}^{1}R^{n}. We show that there exist families of natural projectors, depending on real parameters, defining new representations of the group GL_{n}(R) in certain vector subspaces of T_{2}^{1}R^{n}. MSC 2000: 15A72, 20C33, 20G05, 53A55 Keywords: tensor space of type (r, s), symmetrization, alternation, trace operation, natural projector, tensor space decomposition

1. Introduction

In this paper we give basic definitions and prove basic results of natural projector theory in tensor spaces over the field or real numbers R. The tensor space of type (r, s) over the vector space R^{n} = R × R × ⋯ × R (n factors R) is denoted by T_{s}^{r}R^{n} = R^{n} ⊗ R^{n} ⊗ ⋯ ⊗ R^{n} ⊗ R^{n∗} ⊗ R^{n∗} ⊗ ⋯ ⊗ R^{n∗} (r factors R^{n}, s factors the dual vector space R^{n∗}). We always suppose n ≥ 2. R^{n} is considered with the canonical left action of the general linear group GL_{n}(R), and the tensor space T_{s}^{r}R^{n} is endowed with the induced (tensor) action. Since our discussions are GL_{n}(R)-invariant, the results apply in the well-known sense to any real, n-dimensional vector space E, and to the tensor space T_{s}^{r}E of type (r, s) over E.

∗Research supported by Grants CEZ:J10/98:192400002 and VS 96003 Global Analysis of the Czech Ministry of Education, Youth and Sports, Grant 201/98/0853 of the Czech Grant Agency, and by the Silesian University in Opava.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

We wish to describe a method allowing us to find all GL_{n}(R)-invariant vector subspaces of the vector space T_{s}^{r}R^{n}; indeed, this is equivalent to finding all GL_{n}(R)-equivariant projectors P : T_{s}^{r}R^{n} → T_{s}^{r}R^{n}. In accordance with the terminology of the differential invariant theory, GL_{n}(R)-equivariant projectors are also called natural.

This method complements our previous results on decompositions of tensor spaces, which are not based on the group representation theory (see [4, 5]). It can be applied effectively for any concrete r and s. However, a general formula for the decomposition has not been found.

It seems that the idea to apply the theory of projectors to the problem of decomposing a tensor space of type (r, 0), or (0, s) into its primitive components belongs to H. Weyl [7]. However, this idea has never been developed to a complete theory, or used to an analysis of concrete cases. Later, the same author gives preference of the group representation theory over the ideas of the pure projector theory [6]; a standard restrictive assumption in this approach is usually applied from the very beginning, namely the assumption that the representation space is a vector space over an algebraically closed field.

For basic ideas and generalities on natural projectors in tensor spaces we refer to Krupka (see [3], Sections 4.4 and 7.3).

Let us now recall briefly main concepts. A tensor t ∈ T_{s}^{r}R^{n} is said to be invariant, if g·t = t for all g ∈ GL_{n}(R). A theorem of Gurevich says that an invariant tensor of type (r, s), where r ≠ s, is always the zero tensor, and, if r = s, an invariant tensor is always a linear combination ∑ c_{σ}δ_{i_{σ(1)}}^{j_{1}}δ_{i_{σ(2)}}^{j_{2}} ⋯ δ_{i_{σ(N)}}^{j_{N}} of products of r factors of the Kronecker δ-tensor, where c_{σ} ∈ R, and σ runs through all permutations of the set {1, 2, …, r} (see [1]). Consider a real, N-dimensional vector space E endowed with a left action of GL_{n}(R). A linear mapping F : E → E is called GL_{n}(R)-equivariant, or natural, if F (g · x) = g · F (x) for all x ∈ E and all g ∈ GL_{n}(R). It is a simple observation that F is natural if and only if its components form an invariant tensor [3]. A natural linear mapping P : E → E which is a projector, i.e., satisfies the projector equation P^{2} = P, is called a natural projector.

In Section 2 we collect standard definitions and facts of the theory of projectors in a vector space (see e.g. [2]). Section 3 is devoted to natural linear operators in a vector space endowed with a left action of GL_{n}(R). In Section 4 we introduce natural projectors in tensor spaces and related concepts such as natural projector equations, decomposability, reducibility, and primitivity. Section 5 is concerned with the trace decomposition theory; it is shown that the trace decomposition of a tensor is related to a natural projector determined uniquely by certain conditions. Finally, in Section 6 we describe all natural projectors in the tensor space T_{2}^{1}R^{n}.

It should be pointed out that the method of natural projectors allows us to treat in a unique way the case of tensors of type (r, s), where not necessarily r =0, or s=0. In this sense the natural projector theory represents a generalization of the classical Young–Kronecker decomposition theory (see e.g. [6]), as well as of the trace decomposition theory [4, 5].

2. Projectors

This introductory section contains a brief formulation of standard results of the projector theory in a finite-dimensional, real vector space E (see e.g. [2]).

Let E^{∗} be the dual of E, and let E × E^{∗} ∋ (x, y) → y(x) = ⟨x, y⟩ ∈ R be the natural pairing. The dual A^{∗} : E^{∗} → E^{∗} of a linear mapping A : E → E is defined by the condition ⟨Ax, y⟩ = ⟨x, A^{∗}y⟩ for all x ∈ E, y ∈ E^{∗}. If A, B : E → E are two linear mappings, then (AB)^{∗} = B^{∗}A^{∗},

A linear operator P : E → E is said to be a projector, if P^{2} = P. Clearly, the zero mapping 0, and the identity mapping id_{E}, are projectors.

Lemma 1. Let E be a finite-dimensional, real vector space.

(a) A projector P : E → E defines the direct sum decomposition E = ker P ⊕ im P.

(b) A linear mapping P : E → E is a projector if and only if id_{E} − P is a projector.

(c) If P : E → E is a projector, then Q = αP, where α ∈ R, is a projector if and only if α = 0, 1.

(d) Let P, Q : E → E be two projectors such that im P = im Q = F. Then there exists a unique linear isomorphism U : F → F such that P = U ∘ Q.

Let u^{∗} : E^{∗} → E^{∗} denote the dual of a linear mapping u : E → E. We say that two projectors P, Q : E → E are orthogonal, if ⟨P x, Q^{∗}y⟩ = 0 and ⟨Qx, P^{∗}y⟩ = 0 for all x ∈ E, y ∈ E^{∗}. Obviously, P and Q are orthogonal if and only if QP = 0 and P Q = 0. For every projector P, the projectors P and id_{E} − P are orthogonal.

Lemma 2. Let P, Q : E → E be projectors.

(a) P + Q is a projector if and only if P and Q are orthogonal.

(b) P − Q is a projector if and only if P Q = QP = Q.

(c) If P and Q commute, P Q − QP = 0, then R = P Q = QP is a projector, and im R = im P ∩ im Q.

(d) ker P = im (id − P).

Remark 1. If P + Q is a projector, then condition (a) implies P Q = QP = 0 hence by (c), im P ∩ im Q = {0}. Thus im (P + Q) = im P + im Q is the direct sum of its subspaces im P and im Q.

Remark 2. If P − Q is a projector, condition (b) together with (c) imply that im Q ⊂ im P.

3. Natural linear operators in tensor spaces

Let E be a finite-dimensional, real vector space, endowed with a left action of the general linear group GL_{n}(R), denoted multiplicatively. A linear operator F : E → E is said to be GL_{n}(R)-equivariant, or natural, if F (A·x) = A·F (x) for every x ∈ E and every A ∈ GL_{n}(R). The vector space of natural linear operators on E is denoted N E.

The kernel and the image of a natural linear operator F : E → E are GL_{n}(R)-invariant vector subspaces of E.

Our aim in this section is to study natural linear operators in the tensor space T_{s}^{r}R^{n}. If the canonical basis of R^{n} is denoted by e_{i}, and e^{i} is the dual basis of R^{n∗}, then any tensor t ∈ T_{s}^{r}R^{n} is uniquely expressible in the form

t = t_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}}e_{i_{1}} ⊗ e_{i_{2}} ⊗ ⋯ ⊗ e_{i_{r}} ⊗ e^{j_{1}} ⊗ e^{j_{2}} ⊗ ⋯ ⊗ e^{j_{s}},  (3.1)

where the real numbers t = t_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} are the components of t. We usually write t = t_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}}.

Let (A, x) → x̄ = A · x be the canonical left action of GL_{n}(R) on R^{n}; in the canonical basis of R^{n}, x̄^{i} = A_{j}^{i}x^{j}, where A = A_{j}^{i}. If B = A^{−1}, B = B_{j}^{i}, the tensor action of GL_{n}(R) on T_{s}^{r}R^{n} is given by

t̄= A · t = t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}}e_{i_{1}} ⊗ e_{i_{2}} ⊗ ⋯ ⊗ e_{i_{r}} ⊗ e^{j_{1}} ⊗ e^{j_{2}} ⊗ ⋯ ⊗ e^{j_{s}},  (3.2)

where

t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = A_{k_{1}}^{i_{1}}A_{k_{2}}^{i_{2}} ⋯ A_{k_{r}}^{i_{r}}B_{j_{1}}^{l_{1}}B_{j_{2}}^{l_{2}} ⋯ B_{j_{s}}^{l_{s}}t_{l_{1}l_{2}···l_{s}}^{k_{1}k_{2}···k_{r}}.  (3.3)

A tensor t ∈ T_{s}^{r}R^{n} is said to be invariant, if A · t = t for all A ∈ GL_{n}(R). The following theorem describes all invariant tensors (see [1], and [3]).

Let S_{r} denote the group of permutations σ of the set {1, 2, …, r}.

Lemma 3. Let t ∈ T_{s}^{r}R^{n}.

(a) Assume that r ≠ s. Then t is invariant if and only if t = 0.

(b) Assume that r = s. Then t is invariant if and only if

t_{j_{1}j_{2}···j_{r}}^{i_{1}i_{2}···i_{r}} = ∑_{σ∈S_{r}} a^{σ}δ_{j_{σ(1)}}^{i_{1}}δ_{j_{σ(2)}}^{i_{2}} ⋯ δ_{j_{σ(r)}}^{i_{r}}  (3.4)

for some a^{σ} ∈ R.

Invariant tensors in T_{r}^{r}R^{n} form a real vector space. This vector space is spanned by the invariant tensors

\begin{matrix}E_{σ} = δ_{j_{σ(1)}}^{i_{1}}δ_{j_{σ(2)}}^{i_{2}} ⋯ δ_{j_{σ(r)}}^{i_{r}}e_{i_{1}} ⊗ e_{i_{2}} ⊗ ⋯ ⊗ e_{i_{r}} ⊗ e^{j_{1}} ⊗ e^{j_{2}} ⊗ ⋯ ⊗ e^{j_{r}} \\ = e_{j_{σ(1)}} ⊗ e_{j_{σ(2)}} ⊗ ⋯ ⊗ e_{j_{σ(r)}} ⊗ e^{j_{1}} ⊗ e^{j_{2}} ⊗ ⋯ ⊗ e^{j_{r}}. \end{matrix}  (3.5)

Note that any invariant tensor can be expressed, instead of (3.4), by

t = ∑_{σ∈S_{r}} a^{σ}E_{σ}.  (3.6)

Now we apply Lemma 3 to natural linear mappings F : T_{s}^{r}R^{n} → T_{q}^{p}R^{n}. We have the following simple observation ([3], Section 4.4).

Lemma 4. Let F : T_{s}^{r}R^{n} → T_{q}^{p}R^{n} be a linear mapping,

t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = F_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{p}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{q}}t_{k_{1}k_{2}···k_{q}}^{l_{1}l_{2}···l_{p}}  (3.7)

its expression relative to the canonical basis of R^{n}. F is natural if and only if its components F_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{p}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{q}} are components of an invariant tensor.

If F is identified with a tensor, F becomes an element of the tensor space T_{s+p}^{r+q}R^{n}. Thus by Lemma 3, a nontrivial natural linear mapping F : T_{s}^{r}R^{n} → T_{q}^{p}R^{n} exists if and only if r + q = s + p.

Let us discuss the case p = r, q = s. Then by Lemma 3 (b), F has an expression

F_{j_{1}j_{2}···j_{s} j_{s+1}j_{s+2}···j_{s+r}}^{i_{1}i_{2}···i_{r} i_{r+1}i_{r+2}···i_{r+s}} = ∑_{σ∈S_{r+s}} a_{σ}δ_{j_{σ(1)}}^{i_{1}}δ_{j_{σ(2)}}^{i_{2}} ⋯ δ_{j_{σ(r)}}^{i_{r}}δ_{j_{σ(r+1)}}^{i_{r+1}}δ_{j_{σ(r+2)}}^{i_{r+2}} ⋯ δ_{j_{σ(r+s)}}^{i_{r+s}},  (3.8)

where a_{σ} ∈ R. Clearly, the same is expressed by the equation

t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = ∑_{μ∈S_{r}, ν∈S_{s}} a_{σ}t_{k_{ν(1)}k_{ν(2)}···k_{ν(s)}}^{l_{μ(1)}l_{μ(2)}···l_{μ(r)}} + τ_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}},  (3.9)

where the summation takes place through σ ∈ S_{r+s} of the form of the product of two permutations σ = μν, ν ∈ S_{r}, μ ∈ S_{s} and τ_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} contains all the remaining terms. Note that each term in τ_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} contains at least as one factor the Kronecker δ-tensor multiplied by an expression obtained from t_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} by the trace operation in one superscript and one subscript.

Since F_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{p}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{q}} are components of an invariant tensor, F can also be expressed by means of (3.6) as

F = ∑_{σ∈S_{r+s}} a^{σ}E_{σ}.  (3.10)

If F, G : T_{s}^{r}R^{n} → T_{s}^{r}R^{n} are two natural linear operators, given in components by

t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = F_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{r}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{s}}t_{k_{1}k_{2}···k_{s}}^{l_{1}l_{2}···l_{r}}, t̄_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = G_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{r}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{s}}t_{k_{1}k_{2}···k_{s}}^{l_{1}l_{2}···l_{r}}  (3.11)

then the composed natural linear operator is given by

t^{=}_{j_{1}j_{2}···j_{s}}^{i_{1}i_{2}···i_{r}} = G_{j_{1}j_{2}···j_{s} l_{1}l_{2}···l_{r}}^{i_{1}i_{2}···i_{r} k_{1}k_{2}···k_{s}}F_{k_{1}k_{2}···k_{s} q_{1}q_{2}···q_{r}}^{l_{1}l_{2}···l_{r} p_{1}p_{2}···p_{s}}t_{p_{1}p_{2}···p_{s}}^{q_{1}q_{2}···q_{r}}.  (3.12)

To obtain an explicit formula, one should substitute from (3.8) into (3.12); indeed, this cannot be done effectively in general, but in every concrete case.

4. Natural projectors in tensor spaces

Let E be a finite-dimensional, real vector space, endowed with a left action of the general linear group GL_{n}(R). By a natural projector on E we mean a natural linear operator F : E → E which is a projector. A natural linear operator F is a natural projector if and only if it satisfies the projector equation F^{2} = F. The projector equation represents a system of quadratic equations for the components of F.

If P : E → E is a natural projector, then both vector subspaces im P, ker P of E are GL_{n}(R)-invariant ([2], § 43).

A natural projector P : E → E is said to be decomposable, if there exist a natural projector Q ≠ 0, P and a natural projector R, such that P = Q + R. In this case Q and R are orthogonal (Lemma 2 (a)). A natural projector which is not decomposable is called indecomposable.

P is said to be reducible, if there exists a natural projector Q ≠ 0 such that im Q ⊂ im P and im Q ≠ im P. If P is not reducible, it is called irreducible, or primitive.

Remark 3. Examples show that there exist reducible natural projectors which are not decomposable. Consider the family P_{λ} of natural linear operators in T_{2}^{1}R^{n} defined by the equations

t̄_{jk}^{i} = δ_{k}^{i}t_{pj}^{p} + λδ_{k}^{i}(−nt_{pj}^{p} + t_{jp}^{p}).  (4.1)

One can easily verify that (4.1) consists of natural projectors. Indeed, contracting (4.1) we obtain t̄_{pj}^{p} = t_{pj}^{p} + λ(−nt_{pj}^{p} + t_{jp}^{p}), t̄_{jp}^{p} = nt_{pj}^{p} + λn(−nt_{pj}^{p} + t_{jp}^{p}), and then

t^{=}_{jk}^{i} = δ_{k}^{i}t̄_{pj}^{p} + λδ_{k}^{i}(−nt̄_{pj}^{p} + t̄_{jp}^{p})

= δ_{k}^{i}(t_{pj}^{p} + λ(−nt_{pj}^{p} + t_{jp}^{p})) − λnδ_{k}^{i}(t_{pj}^{p} + λ(−nt_{pj}^{p} + t_{jp}^{p}))

+λδ_{k}^{i}(nt_{pj}^{p} + λn(−nt_{pj}^{p} + t_{jp}^{p}))
= δ_{k}^{i}t_{pj}^{p} − δ_{k}^{i}λnt_{pj}^{p} + δ_{k}^{i}λt_{jp}^{p} = δ_{k}^{i}t_{pj}^{p} + λδ_{k}^{i}(−nt_{pj}^{p} + t_{jp}^{p}) = t̄_{jk}^{i}

verifying the projector equations P_{λ}^{2} = P_{λ}. Note that the family (4.1) includes the natural projector t̄_{jk}^{i} = δ_{k}^{i}t_{pj}^{p}, and the natural projector t̄_{jk}^{i} = (1/n)δ_{k}^{i}t_{jp}^{p} defined by taking λ = 1/n. The family λδ_{k}^{i}(−nt_{pj}^{p} + t_{jp}^{p} in (4.1) does not consist of projectors, because λ serves as a multiplicative parameter, and two non-zero projectors cannot differ by a factor different from 1. Indeed, writing t̄_{qr}^{p} = λδ_{r}^{p}(−nt_{sq}^{s} + t_{qs}^{s}), we get t̄_{pj}^{p} = λ(−nt_{sj}^{s} + t_{js}^{s}), t̄_{jp}^{p} = λn(−nt_{sj}^{s} + t_{js}^{s})

hence t^{=}_{jk}^{i} = λδ_{k}^{i}(−nt̄_{pj}^{p} +t̄_{jp}^{p}) = −lnδ_{k}^{i}t̄_{jp}^{p}+λδ_{k}^{i}t̄_{jp}^{p} = −λ^{2}nδ_{k}^{i}(−nt_{sj}^{s} +t_{js}^{s})+λ^{2}nδ_{k}^{i}(−nt_{sj}^{s} +t_{js}^{s}) =

0 ≠ t̄_{jk}^{i}.

From now on we consider natural projectors on a tensor space T_{s}^{r}R^{n}.

Theorem 1. Let P : T_{s}^{r}R^{n} → T_{s}^{r}R^{n} be a natural projector.

(a) P is decomposable if and only if there exists a natural projector Q ≠ 0, P such that

P Q = Q, QP = Q.  (4.2)

(b) P is reducible if and only if there exists a natural projector Q ≠ 0, P such that

P Q = Q, im Q ≠ im P.  (4.3)

Proof. (a) If P is decomposable, we have two natural projectors Q and R such that R = P −Q and QR = 0, RQ = 0 (Lemma 2 (a)). Thus, Q(P −Q) = (P −Q)Q = 0, i.e., QP = P Q = Q. Conversely, assume that we have a natural projector Q satisfying (4.2). Define R = P − Q; R is a natural linear operator (Lemma 3, Lemma 4), and R^{2} = P − P Q − QP + Q = P − Q − Q + Q = P − Q = R as required.

(b) Let P be reducible. Then there exists a natural projector Q ≠ 0 such that im Q ⊂ im P and im Q ≠ im P. Thus, to any t ∈ T_{s}^{r}R^{n} there exists t′ ∈ T_{s}^{r}R^{n} such that Qt = P t′ = P (P t′) = P Qt hence P Q = Q. Conversely, assume that we have a natural projector Q ≠ 0 satisfying (4.3). Then im Q = Q(T_{s}^{r}R^{n}) = P (Q(T_{s}^{r}R^{n})) ⊂ P (T_{s}^{r}R^{n}) = im P as required. Equations from Theorem 1 (a) for a projector Q

P Q = Q, QP = Q, Q^{2} = Q  (4.4)

are equivalent with the equations

P QP = Q, Q^{2} = Q.  (4.5)

Indeed, (4.4) implies (4.5), and vice versa: QP = P QP P = P QP = Q, P Q = P P QP = P QP = Q. Each of the systems (4.4) and (4.5) is called the decomposability equation of P. Equation P Q = Q from Theorem 1 (b) is called the reducibility equation.

Now we study indecomposability, and primitivity.

Theorem 2. Let P : T_{s}^{r}R^{n} → T_{s}^{r}R^{n} be a natural projector.

(a) P is indecomposable if and only if the decomposability equation of P has exactly one nontrivial solution, Q = P.

(b) P is primitive if and only if the reducibility equation of P has no nontrivial solution.

Proof. Both assertions are immediate consequences of Theorem 1.

(a) If P is indecomposable, there is no Q ≠ 0, P such that P Q = Q, QP = Q, which means that the decomposability equations have only one nontrivial solution, Q = P. The converse is obvious.

(b) If P is primitive, then by definition, (4.3) has only the trivial solution, and vice versa.

Now we consider properties of primitive natural projectors.

Theorem 3.

(a) Any two different primitive natural projectors in T_{s}^{r}R^{n} are orthogonal.

(b) The number of different nontrivial natural projectors in T_{s}^{r}R^{n} is finite.

(c) The sum of any two primitive natural projectors is a natural projector.

(d) Let M be the number of different nontrivial primitive natural projectors in T_{s}^{r}R^{n}. If a natural projector in T_{s}^{r}R^{n} admits a decomposition P = p_{1} + p_{2} + ⋯ + p_{K}, where p_{1}, p_{2},…, p_{K} are primitive natural projectors, then K ≤ M, the primitive natural projectors p_{1}, p_{2},…, p_{K} are mutually different, and this decomposition is unique.

(e) The identity natural projector id : T_{s}^{r}R^{n} → T_{s}^{r}R^{n} admits the decomposition

id = p_{1} + p_{2} + ⋯ + p_{M}  (4.6)

where {p_{1}, p_{2}, …, p_{M}} is the set of nonzero primitive natural projectors.

Proof. (a) If P_{1}, P_{2} are two different primitive natural projectors, then im P_{1}P_{2} = im P_{2}P_{1} = 0 hence P_{1}P_{2} = P_{2}P_{1} = 0.

(b) Since dim T_{s}^{r}R^{n} is finite, this assertion follows from (a).

(c) By (a), any two different primitive natural projectors p_{1}, p_{2} are orthogonal. Thus, by Lemma 2 (a), p_{1}+p_{2} is always a projector; p_{1}+p_{2} is obviously a natural projector (Lemma 4).

(d) Assume that P = p_{1} + p_{2} + ⋯ + p_{K} = q_{1} + q_{2} + ⋯ + q_{L}. Then by orthogonality, p_{l}^{2} = p_{l} = p_{l}(q_{1} + q_{2} + ⋯ + q_{L}), where at most one term on the right is nonzero. But p_{l} ≠ 0 hence exactly one term on the right, say p_{l}q_{k}, is nonzero, and is equal to p_{l}, i.e., p_{l} = p_{l}q_{k} = q_{k}p_{l}. Since different primitive projectors are orthogonal (see (a)), we have q_{k} = p_{l}. In particular, the two sums p_{1} + p_{2} + ⋯ + p_{K}, q_{1} + q_{2} + ⋯ + q_{L} may differ only by the order of the summation.

(e) If P = p_{1} + p_{2} + ⋯ + p_{M} ≠ id, we have a nonzero natural projector Q = id − P, which is a contradiction with maximality of the set {p_{1}, p_{2}, …, p_{M}}.

5. The trace decomposition

For basic notions of the trace decomposition theory as used in this section, we refer to [4], [5]. The following assertion can be used when calculating the trace decomposition of concrete tensor spaces.

Theorem 4. Let r, s ≥ 1. There exists a unique natural linear operator Q : T_{s}^{r}R^{n} → T_{s}^{r}R^{n} satisfying the following two conditions:

1. Qt is traceless for every t ∈ T_{s}^{r}R^{n}.

2. (id − Q)t = t − Qt is δ-generated for every t ∈ T_{s}^{r}R^{n}. Q is a natural projector.

Proof. Existence and uniqueness of Q follows from the decomposition t = Qt + (id − Q)t, and from the trace decomposition theorem. We prove that Q is a projector. By hypothesis, Qt is traceless for every t ∈ T_{s}^{r}R^{n}, hence Q^{2}t = Q(Qt) is also traceless for every t. Similarly, since t − Qt is δ-generated for every t ∈ T_{s}^{r}R^{n}, the formula

(id − Q^{2})t = (id − Q + Q − Q^{2})t = (id − Q)t + (id − Q)Qt  (5.1)

shows that (id−Q^{2})t must also be δ-generated. Since t = Q^{2}t+(id−Q^{2})t, then by uniqueness, Q^{2} = Q.

In a concrete case, the natural projector Q can be determined from the conditions (1) and (2) of Theorem 4. Clearly, given Q, the trace decomposition of a tensor t ∈ T_{s}^{r}R^{n} is obtained by the formula

t = Qt + (id − Q)t.  (5.2)

6. Natural projectors in R^{n} ⊗ R^{n∗} ⊗ R^{n∗}

As an application of the natural projector theory, we find the complete list of natural projectors in the space of tensors of type (1,2) T_{2}^{1}R^{n}. Since our discussions are GL_{n}(R)-invariant, the results apply in the well-known sense to any real, finite-dimensional vector space E, and to the tensor space of type (1,2) over E.

First let us describe natural linear operators in T_{2}^{1}R^{n}. Using the canonical basis e_{i} of R^{n} and the dual basis e^{j} of R^{n∗}, we usually express a tensor t ∈ T_{2}^{1}R^{n} in terms of its components as t = t_{jk}^{i}e_{i} ⊗e^{j} ⊗e^{k}, and we write t = t_{jk}^{i}. If P : T_{2}^{1}R^{n} → T_{2}^{1}R^{n} is a linear operator, we write P = P_{jk p}^{i qr}, where P_{jk p}^{i qr} are the components of P, and the indices i, j, k, p, q, r run through the set {1, 2, …, n}. The equations of P are usually written in the form t̄_{jk}^{i} = P_{jk p}^{i qr}t_{qr}^{p}. P is natural if and only if

P_{jk p}^{i qr} = aδ_{j}^{i}δ_{k}^{q}δ_{p}^{r} + bδ_{j}^{i}δ_{p}^{q}δ_{k}^{r} + cδ_{k}^{i}δ_{p}^{q}δ_{j}^{r} + dδ_{k}^{i}δ_{j}^{q}δ_{p}^{r} + eδ_{p}^{i}δ_{j}^{q}δ_{k}^{r} + fδ_{p}^{i}δ_{k}^{q}δ_{j}^{r},  (6.1)

where a, b, c, d, e, f are some real numbers. In view of (6.1), we also write

P = (a, b, c, d, e, f).  (6.2)

We denote by N(T_{2}^{1}R^{n}) the real vector space of natural linear operators P : T_{2}^{1}R^{n} → T_{2}^{1}R^{n}; by (6.1), dim N (T_{2}^{1}R^{n}) = 6.

We find the composition law for the natural linear operators. Consider a natural linear operator (6.1), and another natural linear operator Q = Q_{bc p}^{a qr}, where

Q_{bc p}^{a qr} = a′δ_{b}^{a}δ_{c}^{q}δ_{p}^{r} + b′δ_{b}^{a}δ_{p}^{q}δ_{c}^{r} + c′δ_{c}^{a}δ_{p}^{q}δ_{b}^{r} + d′δ_{c}^{a}δ_{b}^{q}δ_{p}^{r} + e′δ_{p}^{a}δ_{b}^{q}δ_{c}^{r} + f′δ_{p}^{a}δ_{c}^{q}δ_{b}^{r}.  (6.3)

Lemma 6. The composed natural linear operator R = P Q = R_{jk p}^{i qr} is expressed by

R_{jk p}^{i qr} = a′′δ_{j}^{i}δ_{k}^{q}δ_{p}^{r} + b′′δ_{j}^{i}δ_{p}^{q}δ_{k}^{r} + c′′δ_{k}^{i}δ_{p}^{q}δ_{j}^{r} + d′′δ_{k}^{i}δ_{j}^{q}δ_{p}^{r} + e′′δ_{p}^{i}δ_{j}^{q}δ_{k}^{r} + f′′δ_{p}^{i}δ_{k}^{q}δ_{j}^{r},  (6.4)

where

\begin{matrix}a′′ = a′a + nd′a + e′a + na′b + f′b + d′b + a′e + d′f, \\ b′′ = b′a + nc′a + f′a + nb′b + c′b + e′b + b′e + c′f, \\ c′′ = nb′c + c′c + e′c + b′d + nc′d + f′d + c′e + b′f, \\ d′′ = na′c + d′c + f′c + a′d + nd′d + e′d + d′e + a′f, \\ e′′ = e′e + f′f, \\ f′′ = f′e + e′f. \end{matrix}  (6.5)

Proof. Since for any t ∈ T_{2}^{1}R^{n}, t = t_{qr}^{p}, Rt = ¯ \overline{t}_{jk}^{i} = P_{jk a}^{i bc} t̄_{bc}^{a} = P_{jk a}^{i bc}Q_{bc p}^{a qr}t_{qr}^{p} = R_{jk}^{i} _{p}^{qr}t_{qr}^{p}, the coefficients R_{jk p}^{i qr} are obtained from the formula

R_{jk p}^{i qr} = P_{jk a}^{i bc} Q_{bc p}^{a qr}.  (6.6)

Now we derive the equations for natural projectors in T_{2}^{1}R^{n}.

Lemma 7. A natural linear operator P : T_{2}^{1}R^{n} → T_{2}^{1}R^{n} expressed by (6.1), is a natural projector if and only if

\begin{matrix}a^{2} + (nb + nd + 2e − 1) a + bd + (b + d)f = 0, \\ nb^{2} + (a + c + 2e − 1) b + nca + (a + c)f = 0, \\ c^{2} + (nb + nd + 2e − 1) c + bd + (b + d)f = 0, \\ nd^{2} + (a + c + 2e − 1) d + nac + (a + c)f = 0, \\ e = e^{2} + f^{2}, \\ f = 2ef. \end{matrix}  (6.7)

Proof. The components of P satisfy the projector equation P_{jk u}^{i vw} P_{vw p}^{u qr} = P_{jk p}^{i qr}, which can be obtained by substituting Q = P and R = P in (6.5).

Equations (6.7) are referred to as the natural projector equations. These equations represent a system of six quadratic equations for six unknowns (a, b, c, d, e, f).

Remark 4. If P is a natural projector, then the complementary projector id − P is also natural. Thus, if P (6.1) satisfies (6.7), then id − P also satisfies (6.7). Indeed,

id − P = a′δ_{j}^{i}δ_{k}^{q}δ_{p}^{r} + b′δ_{j}^{i}δ_{p}^{q}δ_{k}^{r} + c′δ_{k}^{i}δ_{p}^{q}δ_{j}^{r} + d′δ_{k}^{i}δ_{j}^{q}δ_{p}^{r} + e′δ_{p}^{i}δ_{j}^{q}δ_{k}^{r} + f′δ_{p}^{i}δ_{k}^{q}δ_{j}^{r},  (6.8)

where

a′ = −a, b′ = −b, c′ = −c, d′ = −d, e′ = 1 − e, f′ = −f.  (6.9)

The transformation (6.9) leaves invariant the system (6.7).

It is easily seen that the formulas

\begin{matrix}a′ = c, b′ = b, c′ = a, d′ = d, e′ = e, f′ = f, \\ a′ = a, b′ = d, c′ = c, d′ = b, e′ = e, f′ = f \end{matrix}  (6.10)

also define invariant transformations of (6.7). Consequently, if (a, b, c, d, e, f) is a natural projector, then also (−a, −b, −c, −d, 1 − e, −f), (c, b, a, d, e, f), (a, d, c, b, e, f), are natural projectors.

We are now in a position to find all solutions of the natural projector equations (6.7). We write these solutions in the form of their equations t̄_{jk}^{i} = P_{jk p}^{i qr} t_{qr}^{p}, i.e., as

t̄_{jk}^{i} = aδ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + et_{jk}^{i} + ft_{kj}^{i}.  (6.11)

Here (a, b, c, d, e, f) are the components (6.2) of a natural projector, expressed by (6.1). Note that the list (A1), (A2), …, (D4) below includes one-, and two-parameter families of natural projectors.

We define

\begin{matrix}A_{1} = nd + d^{2} − n^{2}d^{2}, & A_{2} = −nd + d^{2} − n^{2}d^{2}, \\ A_{3} = n^{2}d^{2} − d^{2} − d, & A_{4} = n^{2}d^{2} − d^{2} + d, \\ B_{1} = n^{2}c^{2} − c^{2} + c, & B_{2} = n^{2}c^{2} − c^{2} − c, \\ C_{1} = 4d + 4d^{2} − 4n^{2}d^{2} + 1, & C_{2} = −4d + 4d^{2} − 4n^{2}d^{2} + 1. \end{matrix}  (6.12)

Theorem 5. The following list contains all natural projectors P : T_{2}^{1}R^{n} → T_{2}^{1}R^{n}:

\begin{matrix}t̄_{jk}^{i} = 0, \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(−δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} − δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}), \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}), \\ t̄_{jk}^{i} = −\frac{1}{n^{2}−1}δ_{j}^{i}t_{ks}^{s} + \frac{n}{n^{2}−1}δ_{j}^{i}t_{sk}^{s} − \frac{1}{n^{2}−1}δ_{k}^{i}t_{sj}^{s} + \frac{n}{n^{2}−1}δ_{k}^{i}t_{js}^{s}, \end{matrix}  (A1)

\begin{matrix}t̄_{jk}^{i} = \frac{−d+\sqrt{A_{1}}}{n} δ_{j}^{i}t_{ks}^{s} + \frac{n+2d−n^{2}d−2\sqrt{A_{1}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} + \frac{−d+\sqrt{A_{1}}}{n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s}, \\ t̄_{jk}^{i} = −\frac{d+\sqrt{A_{1}}}{n} δ_{j}^{i}t_{ks}^{s} + \frac{n+2d−n^{2}d+2\sqrt{A_{1}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} − \frac{d+\sqrt{A_{1}}}{n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s}, \\ d ∈ [0, n/(n^{2} − 1)], \end{matrix}  (A2)

\begin{matrix}t̄_{jk}^{i} = (1 − c − 2n (−nc + \sqrt{B_{1}})) δ_{j}^{i}t_{ks}^{s} + (−nc + \sqrt{B_{1}}) δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} \\ + (−nc + \sqrt{B_{1}}) δ_{k}^{i}t_{js}^{s}, \\ t̄_{jk}^{i} = (1 − c + 2n (nc + \sqrt{B_{1}})) δ_{j}^{i}t_{ks}^{s} − (nc + \sqrt{B_{1}}) δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} \\ − (nc + \sqrt{B_{1}}) δ_{k}^{i}t_{js}^{s}, \\ c ∈ (−∞, −1/(n^{2} − 1)] ∪ [0, ∞), \end{matrix}  (A3)

\begin{matrix}t̄_{jk}^{i} = (1 − nb)δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s}, \\ t̄_{jk}^{i} = \frac{d−nd^{2}−cd}{d+nc}δ_{j}^{i}t_{ks}^{s} + \frac{c−c^{2}−ncd}{d+nc}δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s}, d + nc ≠ 0, \\ t̄_{jk}^{i} = −nbδ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} − \frac{1}{n^{2}−1}δ_{k}^{i}t_{sj}^{s} + \frac{n}{n^{2}−1}δ_{k}^{i}t_{js}^{s}, \end{matrix}  (A4)

\begin{matrix}t̄_{jk}^{i} = t_{jk}^{i}, \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} − δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} − δ_{k}^{i}t_{js}^{s}) + t_{jk}^{i}, \\ t̄_{jk}^{i} = −\frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}) + t_{jk}^{i}, \\ t̄_{jk}^{i} = \frac{1}{n^{2}−1}(δ_{j}^{i}t_{ks}^{s} − nδ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} − nδ_{k}^{i}t_{js}^{s}) + t_{jk}^{i}, \end{matrix}  (B1)

\begin{matrix}t̄_{jk}^{i} = \frac{−d+\sqrt{A_{2}}}{n} δ_{j}^{i}t_{ks}^{s} + \frac{−n+2d−n^{2}d−2\sqrt{A_{2}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} + \frac{−d+\sqrt{A_{2}}}{n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, \\ t̄_{jk}^{i} = −\frac{d+\sqrt{A_{2}}}{n} δ_{j}^{i}t_{ks}^{s} + \frac{−n+2d−n^{2}d+2\sqrt{A_{2}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} − \frac{d+\sqrt{A_{2}}}{n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, \\ d ∈ [−n/(n^{2} − 1), 0], \end{matrix}  (B2)

\begin{matrix}t̄_{jk}^{i} = (1 − c − 2n (−nc + \sqrt{B_{2}})) δ_{j}^{i}t_{ks}^{s} + (−nc + \sqrt{B_{2}}) δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} \\ + (−nc + \sqrt{B_{2}}) δ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, \\ t̄_{jk}^{i} = (1 − c − 2n (−nc − \sqrt{B_{2}})) δ_{j}^{i}t_{ks}^{s} − (nc + \sqrt{B_{2}}) δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} \\ − (nc + \sqrt{B_{2}}) δ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, \\ c ∈ (−∞, 0] ∪ [1/(n^{2} − 1), ∞), \end{matrix}  (B3)

\begin{matrix}t̄_{jk}^{i} = (1 − nb)δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} + t_{jk}^{i}, \\ t̄_{jk}^{i} = −\frac{d+nd^{2}+cd}{d+nc}δ_{j}^{i}t_{ks}^{s} − \frac{c+c^{2}+ncd}{d+nc}δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, d + nc ≠ 0, \\ t̄_{jk}^{i} = −nbδ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} + \frac{1}{n^{2}−1}δ_{k}^{i}t_{sj}^{s} − \frac{n}{n^{2}−1}δ_{k}^{i}t_{js}^{s} + t_{jk}^{i}, \end{matrix}  (B4)

\begin{matrix}t̄_{jk}^{i} = \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(−δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} − δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}) + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = −\frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}) + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = −\frac{n}{n^{2}−1}δ_{j}^{i}t_{ks}^{s} + \frac{1}{n^{2}−1}δ_{j}^{i}t_{sk}^{s} − \frac{n}{n^{2}−1}δ_{k}^{i}t_{sj}^{s} + \frac{1}{n^{2}−1}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \end{matrix}  (C1)

\begin{matrix}t̄_{jk}^{i} = −\frac{1+2d+\sqrt{C_{1}}}{2n} δ_{j}^{i}t_{ks}^{s} + \frac{2d−n^{2}d+1+\sqrt{C_{1}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} − \frac{1+2d+\sqrt{C_{1}}}{2n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + et_{jk}^{i} + ft_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{−1−2d+\sqrt{C_{1}}}{2n} δ_{j}^{i}t_{ks}^{s} + \frac{2d−n^{2}d+1−\sqrt{C_{1}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} + \frac{−1−2d+\sqrt{C_{1}}}{2n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ d ∈ [−\frac{1}{2}(n + 1), \frac{1}{2}(n + 1)], \end{matrix} (C2)

\begin{matrix}t̄_{jk}^{i} = − (nd + \sqrt{A_{3}}) δ_{j}^{i}t_{ks}^{s} + dδ_{j}^{i}t_{sk}^{s} + (−nd + \sqrt{A_{3}})δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = (−nd + \sqrt{A_{3}})δ_{j}^{i}t_{ks}^{s} + dδ_{j}^{i}t_{sk}^{s} − (nd + \sqrt{A_{3}}) δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ d ∈ (−∞, 0] ∪ [1/(n^{2} − 1), ∞), \end{matrix}  (C3)

\begin{matrix}t̄_{jk}^{i} = − (\frac{1}{2} + nb) δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} − \frac{1}{2(n−1)}δ_{k}^{i}t_{sj}^{s} + \frac{1}{2(n−1)}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = −\frac{c+2nd^{2}+2cd}{2d+2nc+1}δ_{j}^{i}t_{ks}^{s} − \frac{d+2ncd+2c^{2}}{2d+2nc+1} δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \\ 2d + 2nc + 1 ≠ 0, \\ t̄_{jk}^{i} = (\frac{1}{2} − nb) δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} − \frac{1}{2(n−1)}δ_{k}^{i}t_{sj}^{s} − \frac{1}{2(n−1)}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} + \frac{1}{2}t_{kj}^{i}, \end{matrix}  (C4)

\begin{matrix}t̄_{jk}^{i} = \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} − δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} − δ_{k}^{i}t_{js}^{s}) + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{1}{2(n−1)}(δ_{j}^{i}t_{ks}^{s} + δ_{j}^{i}t_{sk}^{s} + δ_{k}^{i}t_{sj}^{s} + δ_{k}^{i}t_{js}^{s}) + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{n}{n^{2}−1}δ_{j}^{i}t_{ks}^{s} − \frac{1}{n^{2}−1}δ_{j}^{i}t_{sk}^{s} + \frac{n}{n^{2}−1}δ_{k}^{i}t_{sj}^{s} − \frac{1}{n^{2}−1}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \end{matrix}  (D1)

\begin{matrix}t̄_{jk}^{i} = \frac{1−2d−\sqrt{C_{2}}}{2n} δ_{j}^{i}t_{ks}^{s} + \frac{2d−n^{2}d−1+\sqrt{C_{2}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} + \frac{1−2d−\sqrt{C_{2}}}{2n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{1−2d+\sqrt{C_{2}}}{2n} δ_{j}^{i}t_{ks}^{s} + \frac{2d−n^{2}d−1−\sqrt{C_{2}}}{n^{2}} δ_{j}^{i}t_{sk}^{s} + \frac{1−2d+\sqrt{C_{2}}}{2n} δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ d ∈ [−1/(n^{2} − 1), 1/(n^{2} − 1)], \end{matrix}  (D2)

\begin{matrix}t̄_{jk}^{i} = − (nd + \sqrt{A_{4}}) δ_{j}^{i}t_{ks}^{s} + dδ_{j}^{i}t_{sk}^{s} + (−nd + \sqrt{A_{4}}) δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = (−nd + \sqrt{A_{4}}) δ_{j}^{i}t_{ks}^{s} + dδ_{j}^{i}t_{sk}^{s} − (nd + \sqrt{A_{4}}) δ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ d ∈ (−∞, −1/(n^{2} − 1)] ∪ [0, ∞), \end{matrix}  (D3)

\begin{matrix}t̄_{jk}^{i} = − (\frac{1}{2} + nb) δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} + \frac{1}{2(n−1)}δ_{k}^{i}t_{sj}^{s} + \frac{1}{2(n−1)}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}t_{kj}^{i}, \\ t̄_{jk}^{i} = \frac{c−2nd^{2}−2cd}{2d+2nc−1}δ_{j}^{i}t_{ks}^{s} + \frac{d−2ncd−2c^{2}}{2d+2nc−1} δ_{j}^{i}t_{sk}^{s} + cδ_{k}^{i}t_{sj}^{s} + dδ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}ft_{kj}^{i}, \\ 2d + 2nc − 1 ≠ 0, \\ t̄_{jk}^{i} = (\frac{1}{2} − nb) δ_{j}^{i}t_{ks}^{s} + bδ_{j}^{i}t_{sk}^{s} + \frac{1}{2(n−1)}δ_{k}^{i}t_{sj}^{s} − \frac{1}{2(n−1)}δ_{k}^{i}t_{js}^{s} + \frac{1}{2}t_{jk}^{i} − \frac{1}{2}ft_{kj}^{i}. \end{matrix}  (D4)

Proof. (6.7) splits into the following 16 cases to be considered separately:

(e, f) = (0, 0), a = c, b = d,  (A1)
(e, f) = (0, 0), a = c, b = −d − \frac{1}{n}(a + c + 2e − 1),  (A2)
(e, f) = (0, 0), a = −c − nb − nd − 2e + 1, b = d,  (A3)
(e, f) = (0, 0), a = −c − nb − nd − 2e + 1, b = −d − \frac{1}{n}(a + c + 2e − 1),  (A4)
(e, f) = (1, 0), a = c, b = d,  (B1)
(e, f) = (1, 0), a = c, b = −d − \frac{1}{n}(a + c + 2e − 1),  (B2)
(e, f) = (1, 0), a = −c − nb − nd − 2e + 1, b = d,  (B3)
(e, f) = (1, 0), a = −c − nb − nd − 2e + 1, b = −d − \frac{1}{n}(a + c + 2e − 1),  (B4)
(e, f) = (\frac{1}{2}, \frac{1}{2}) , a = c, b = d,  (C1)
(e, f) = (\frac{1}{2}, \frac{1}{2}) , a = c, b = −d − \frac{1}{n}(a + c + 2e − 1),  (C2)
(e, f) = (\frac{1}{2}, \frac{1}{2}) , a = −c − nb − nd − 2e + 1, b = d,  (C3)

(e, f) = (\frac{1}{2}, \frac{1}{2}) , a = −c − nb − nd − 2e + 1, b = −d − \frac{1}{n}(a + c + 2e − 1)  (C4)
(e, f) = (\frac{1}{2}, −\frac{1}{2}) , a = c, b = d,  (D1)
(e, f) = (\frac{1}{2}, −\frac{1}{2}) , a = c, b = −d − \frac{1}{n}(a + c + 2e − 1),  (D2)
(e, f) = (\frac{1}{2}, −\frac{1}{2}) , a = −c − nb − nd − 2e + 1, b = d,  (D3)
(e, f) = (\frac{1}{2}, −\frac{1}{2}) , a = −c − nb − nd − 2e + 1, b = −d − \frac{1}{n}(a + c + 2e − 1).  (D4)

Each of these cases is subject to the conditions

\begin{matrix}a^{2} + c^{2} + (nb + nd + 2e − 1)(a + c) + 2bd + 2(b + d)f = 0, \\ nb^{2} + nd^{2} + (a + c + 2e − 1)(b + d) + 2nac + 2(a + c)f = 0. \end{matrix}  (6.13)

To complete the proof, we solve the system (6.13) of two quadratic equations for every of the possibilities (A1), (A2), …, (D4). We get, using MAPLE,

\begin{matrix}(0, 0, 0, 0, 0, 0), \\ (−\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, 0, 0) , \\ (\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2(n−1)}, 0, 0) , \\ (−\frac{1}{n^{2}−1}, \frac{n}{n^{2}−1}, −\frac{1}{n^{2}−1}, \frac{n}{n^{2}−1}, 0, 0) , \end{matrix}  (A1)

\begin{matrix}(\frac{−d+\sqrt{A_{1}}}{n} , \frac{n+2d−n^{2}d−2\sqrt{A_{1}}}{n^{2}} , \frac{−d+\sqrt{A_{1}}}{n} , d, 0, 0) , \\ (−\frac{d+\sqrt{A_{1}}}{n} , \frac{n+2d−n^{2}d+2\sqrt{A_{1}}}{n^{2}} , −\frac{d+\sqrt{A_{1}}}{n} , d, 0, 0) , \\ d ∈ [0, n/(n^{2} − 1)], \end{matrix}  (A2)

\begin{matrix}(1 − c − 2n (−nc + \sqrt{B_{1}}) , −nc + \sqrt{B_{1}}, c, −nc + \sqrt{B_{1}}, 0, 0) , \\ (1 − c + 2n (nc + \sqrt{B_{1}}) , − (nc + \sqrt{B_{1}}) , c, − (nc + \sqrt{B_{1}}) , 0, 0) , \\ c ∈ (−∞, −1/(n^{2} − 1)] ∪ [0, ∞), \end{matrix}  (A3)

\begin{matrix}(1 − nb, b, 0, 0, 0, 0), \\ (\frac{d−nd^{2}−cd}{d+nc}, \frac{c−c^{2}−ncd}{d+nc}, c, d, 0, 0) , d + nc ≠ 0, \\ (−nb, b, −\frac{1}{n^{2}−1}, \frac{n}{n^{2}−1}, 0, 0) , \end{matrix}  (A4)

\begin{matrix}(0, 0, 0, 0, 1, 0), \\ (\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, 1, 0) , \\ (−\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, 1, 0) , \\ (\frac{1}{n^{2}−1}, −\frac{n}{n^{2}−1}, \frac{1}{n^{2}−1}, −\frac{n}{n^{2}−1}, 1, 0) , \end{matrix}  (B1)

\begin{matrix}(\frac{−d+\sqrt{A_{2}}}{n} , \frac{−n+2d−n^{2}d−2\sqrt{A_{2}}}{n^{2}} , \frac{−d+\sqrt{A_{2}}}{n} , d, 0, 0) , \\ (−\frac{d+\sqrt{A_{2}}}{n} , \frac{−n+2d−n^{2}d+2\sqrt{A_{2}}}{n^{2}} , −\frac{d+\sqrt{A_{2}}}{n} , d, 0, 0) , \\ d ∈ [−n/(n^{2} − 1), 0], \end{matrix}  (B2)

\begin{matrix}(1 − c − 2n (−nc + \sqrt{B_{2}}) , −nc + \sqrt{B_{2}}, c, −nc + \sqrt{B_{2}}, 1, 0) , \\ (1 − c − 2n (−nc − \sqrt{B_{2}}) , −nc − \sqrt{B_{2}}, c, −nc − \sqrt{B_{2}}, 1, 0) , \\ c ∈ (−∞, 0] ∪ [1/(n^{2} − 1), ∞), \end{matrix}  (B3)

\begin{matrix}(1 − nb, b, 0, 0, 1, 0), \\ (−\frac{d+nd^{2}+cd}{d+nc}, −\frac{c+c^{2}+ncd}{d+nc}, c, d, 1, 0) , d + nc ≠ 0, \\ (−nb, b, \frac{1}{n^{2}−1}, −\frac{n}{n^{2}−1}, 1, 0) , \end{matrix}  (B4)

\begin{matrix}(0, 0, 0, 0, \frac{1}{2}, \frac{1}{2}) , \\ (−\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2}, \frac{1}{2}) , \\ (−\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2}, \frac{1}{2}) , \\ (−\frac{n}{n^{2}−1}, \frac{1}{n^{2}−1}, −\frac{n}{n^{2}−1}, \frac{1}{n^{2}−1}, \frac{1}{2}, \frac{1}{2}) , \end{matrix}  (C1)

\begin{matrix}(−\frac{1+2d+\sqrt{C_{1}}}{2n} , \frac{2d−n^{2}d+1+\sqrt{C_{1}}}{n^{2}} , −\frac{1+2d+\sqrt{C_{1}}}{2n} , d, \frac{1}{2}, \frac{1}{2}) , \\ (\frac{−1−2d+\sqrt{C_{1}}}{2n} , \frac{2d−n^{2}d+1−\sqrt{C_{1}}}{n^{2}} , \frac{−1−2d+\sqrt{C_{1}}}{2n} , d, \frac{1}{2}, \frac{1}{2}) , \\ d ∈ [−\frac{1}{2}(n + 1), \frac{1}{2}(n + 1)], \end{matrix}  (C2)

\begin{matrix}(−nd − \sqrt{A_{3}}, d, −nd + \sqrt{A_{3}}, d, \frac{1}{2}, \frac{1}{2}) , \\ (−nd + \sqrt{A_{3}}, d, −nd − \sqrt{A_{3}}, d, \frac{1}{2}, \frac{1}{2}) , \\ d ∈ (−∞, 0] ∪ [1/(n^{2} − 1), ∞), \end{matrix}  (C3)

\begin{matrix}(−\frac{1}{2} − nb, b, −\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2}, \frac{1}{2}) , \\ (−\frac{c+2nd^{2}+2cd}{2d+2nc+1}, −\frac{d+2ncd+2c^{2}}{2d+2nc+1} , c, d, \frac{1}{2}, \frac{1}{2}) , 2d + 2nc + 1 ≠ 0, \\ (\frac{1}{2} − nb, b, −\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2}, \frac{1}{2}) , \end{matrix}  (C4)

\begin{matrix}(0, 0, 0, 0, \frac{1}{2}, −\frac{1}{2}) , \\ (\frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2}, −\frac{1}{2}) , \\ (\frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2}, −\frac{1}{2}) , \\ (\frac{n}{n^{2}−1}, −\frac{1}{n^{2}−1}, \frac{n}{n^{2}−1}, −\frac{1}{n^{2}−1}, \frac{1}{2}, −\frac{1}{2}) , \end{matrix}  (D1)

\begin{matrix}(\frac{1−2d−\sqrt{C_{2}}}{2n} , \frac{2d−n^{2}d−1+\sqrt{C_{2}}}{n^{2}} , \frac{1−2d−\sqrt{C_{2}}}{2n} , d, \frac{1}{2}, −\frac{1}{2}) , \\ (\frac{1−2d+\sqrt{C_{2}}}{2n} , \frac{2d−n^{2}d−1−\sqrt{C_{2}}}{n^{2}} , \frac{1−2d+\sqrt{C_{2}}}{2n} , d, \frac{1}{2}, −\frac{1}{2}) , \\ d ∈ [−1/(n^{2} − 1), 1/(n^{2} − 1)], \end{matrix}  (D2)

\begin{matrix}(−nd − \sqrt{A_{4}}, d, −nd + \sqrt{A_{4}}, d, \frac{1}{2}, −\frac{1}{2}) , \\ (−nd + \sqrt{A_{4}}, d, −nd − \sqrt{A_{4}}, d, \frac{1}{2}, −\frac{1}{2}) , \\ d ∈ (−∞, −1/(n^{2} − 1)] ∪ [0, ∞), \end{matrix}  (D3)

\begin{matrix}(−\frac{1}{2} − nb, b, \frac{1}{2(n−1)}, \frac{1}{2(n−1)}, \frac{1}{2}, −\frac{1}{2}) , \\ (\frac{c−2nd^{2}−2cd}{2d+2nc−1}, \frac{d−2ncd−2c^{2}}{2d+2nc−1} , c, d, \frac{1}{2}, −\frac{1}{2}) , 2d + 2nc − 1 ≠ 0, \\ (\frac{1}{2} − nb, b, \frac{1}{2(n−1)}, −\frac{1}{2(n−1)}, \frac{1}{2}, −\frac{1}{2}) . \end{matrix}  (D4)

Now our assertion follows from (6.11).

Remark 5. Note that Theorem 5 gives us a complete answer to the problem of finding all natural projectors in T_{2}^{1}R^{n}. Properties of these natural projectors can be obtained from this list (A1), (A2), …, (D4) by a direct analysis.

References

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[2] Halmos, P. R.: Finite-Dimensional Vector Spaces. Russian translation, GIFM, Moscow 1963. Zbl 0107.01501

[3] Krupka, D.; Janyska, J.: Lectures on Differential Invariants. Brno University, Brno, Czech Republic, 1990. Zbl 0752.53004

[4] Krupka, D.: The trace decomposition problem. Beiträge Algebra Geom. 36 (1995), 303– 315. Zbl 0839.15024

[5] Krupka, D.: The trace decomposition of tensors of type (1,2) and (1,3). In: New Developments in Differential Geometry, L. Tamássy and J. Szenthe (eds.), Proc. Coll. on Diff. Geom., Debrecen, Hungary, 1994; Kluwer, Dordrecht 1996, 243–253. Zbl 0840.15024

[6] Weyl, H.: The Classical Groups, their Invariants and Representations. Second edition, Princeton University Press, Princeton, New Jersey 1946. Zbl 0020.20601

[7] Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publications 1931. Zbl 0041.56804

Received February 23, 2000