Séminaire Lotharingien de Combinatoire 48 (2003), Article B48d

DETERMINATION OF THE STRUCTURE OF ALGEBRAIC

CURVATURE TENSORS BY MEANS OF YOUNG SYMMETRIZERS

BERND FIEDLER

Abstract. For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors. We verify our results by means of the Littlewood–Richardson rule and plethysms. For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS.

1. Introduction

Let T_{r}V be the vector space of the r-times covariant tensors T over a finitedimensional 𝕂-vector space V, 𝕂 = ℝ or 𝕂 = ℂ. Definition 1.1. A tensor T ∈ T_{4}V is called an algebraic curvature tensor iff T has the index commutation symmetry

(1.1) ∀ u, x, y, z ∈ V : T (u, x, y, z) = −T (u, x, z, y) = T (y, z, u, x)

and fulfills the first Bianchi identity

(1.2) ∀ u, x, y, z ∈ V : T (u, x, y, z) + T (u, y, z, x) + T (u, z, x, y) = 0 .

If we consider the coordinates T_{ijkl} of a tensor T ∈ T_{r}V then T is an algebraic curvature tensor iff its coordinates satisfy

(1.3)  T_{ijkl} = −T_{jikl} = −T_{ijlk} = T_{klij}

and

(1.4)  T_{ijkl} + T_{iklj} + T_{iljk} = 0 .

with respect to every basis of V.

We assume that V possesses a fundamental tensor g ∈ T_{2}V (of arbitrary signature) which can be used for raising and lowering of tensor indices. Definition 1.2. Let T ∈ T_{4}V be an algebraic curvature tensor and x ∈ V be a vector with |g(x, x)| = 1. The Jacobi operator J_{T} (x) of T and x is the linear operator J_{T} (x) : V → V , J_{T} (x) : y ↦ J_{T} (x)y that is defined by

(1.5)  ∀ w ∈ V : g(J_{T} (x)y, w) = T (y, x, x, w) .

2000 Mathematics Subject Classification. 53B20, 15A72, 05E10, 16D60, 05-04.

Furthermore we use the following operators γ and α:

Definition 1.3. (1) Let S ∈ T_{2}V be a symmetric tensor of order 2, i.e. the coordinates of S satisfy S_{ij} = S_{ji}. We define a tensor γ(S) ∈ T_{4}V by

(1.6)  γ(S)_{ijkl} := \frac{1}{3} (S_{il}S_{jk} − S_{ik}S_{jl}) .

(2) Let A ∈ T_{2}V be a skew-symmetric tensor of order 2, i.e. the coordinates of A satisfy A_{ji} = −A_{ij}. We define a tensor α(A) ∈ T_{4}V by

(1.7)  α(A)_{ijkl} := \frac{1}{3} (2 A_{ij}A_{kl} + A_{ik}A_{jl} − A_{il}A_{jk}) .

Tensors of type γ(S) and α(A) occur in formulas of so-called Osserman tensors (see [24]).

Definition 1.4. An algebraic curvature tensor T is called spacelike Osserman (respectively timelike Osserman) if the eigenvalues of J_{T} (x) are constant on S^{+}(V) := {x ∈ V | g(x, x) = +1} (respectively S^{−}(V) := {x ∈ V | g(x, x) = −1}).

Since the fundamental tensor g connects an algebraic curvature tensor T with its Jacobi operator J_{T} (x) in (1.5), the Osserman property depends not only on T but also on g. It is known that spacelike Osserman and timelike Osserman are equivalent notions so one simply says Osserman.

If R is the Riemann tensor of a Riemannian manifold (M, g) which is locally a rank one symmetric space or flat, then the eigenvalues of J_{R}(x) are constant on the unit sphere. Osserman [37] wondered if the converse held. This question is known as the Osserman conjecture.

The correctness of the Osserman conjecture has been established for Riemannian manifolds (M, g) in all dimensions ≠ 8, 16 (see [7, 36]) and for Lorentzian manifolds (M, g) in all dimensions (see [1, 20]). The situation is much more complicated in the case of a pseudo-Riemannian metric with signature (p, q), p, q ≥ 2. It is known that there exist pseudo-Riemannian Osserman manifolds which are not locally homogeneous [2, 6, 22]. Furthermore there are counter examples to the conjecture if the manifold (M, g) has signature (2, 2) and if the Jacobi operator is not diagonalizable [2, 3]. (See also the examples in [21, 38]). Thus the question Osserman raised has a negative answer in the higher signature setting. More extensive bibliographies in the subject can be found in the books [26] by P. B. Gilkey and [23] by E. García–Rio, D. N. Kupeli and R. Vázquez–Lorenzo.

Examples of Osserman algebraic curvature tensors can be constructed in the following way. Let {C_{i}}_{i=1}^{r} be a finite set of real, skew-symmetric (dimV × dimV)- matrices that satisfy the Clifford commutation relations

(1.8)  C_{i} · C_{j} + C_{j} · C_{i} = −2 δ_{ij} .

We define

(1.9) T_{C}(x, y)z := g(y, C · z) C · x − g(x, C · z) C · y − 2 g(x, C · y) C · z ,
(1.10) T_{0}(x, y)z := g(y, z) x − g(x, z) y , x, y, z ∈ V ,
(1.11)  T := λ_{0} T_{0} + λ_{1} T_{C_{1}} + … + λ_{r} T_{C_{r}} .

Now if g is positive definite, then T is an algebraic curvature tensor that is Osserman. (See [25, 27, 24] for more details.)

T_{0} and T_{C} can be expressed by means of γ and α, respectively, since

g(T_{0}(x, y) z, w) = g(y, z) g(x, w) − g(x, z) g(y, w)
= (g_{jk}g_{il} − g_{ik}g_{jl}) x^{i}y^{j}z^{k}w^{l}
= 3 γ(g)_{ijkl} x^{i}y^{j}z^{k}w^{l}

and

g(T_{C}(x, y) z, w) = g(y, C · z) g(C · x, w) − g(x, C · z) g(C · y, w) −
2 g(x, C · y) g(C · z, w)
= g_{js}y^{j}C^{s}_{k}z^{k} g_{tl}C^{t}_{i}x^{i}w^{l} − g_{is}x^{i}C^{s}_{k}z^{k} g_{tl}C^{t}_{j}y^{j}w^{l} −
2 g_{is}x^{i}C^{s}_{j}y^{j} g_{tl}C^{t}_{k}z^{k}w^{l}
= {C_{jk}C_{li} − C_{ik}C_{lj} − 2 C_{ij}C_{lk}} x^{i}y^{j}z^{k}w^{l}
= {−C_{jk}C_{il} + C_{ik}C_{jl} + 2 C_{ij}C_{kl}} x^{i}y^{j}z^{k}w^{l}
= 3 α(C)_{ijkl} x^{i}y^{j}z^{k}w^{l} .

Thus we obtain

(1.12)  T = 3 λ_{0} γ(g) + 3 ∑_{i=1}^{r} λ_{i} α(C_{i})

from (1.11). It was conjectured that any algebraic Osserman curvature tensor has a structure given in (1.11). This is true for all known examples of Osserman tensors in the case of a positive definite metric g (see [24]). But for indefinite metrics there exist counter examples (see Remark 6.7).

In the present paper we show that every algebraic curvature tensor T has a representation

(1.13)  T = ∑_{S∈S} ϵ_{S} γ(S) + ∑_{A∈A} ϵ_{A} α(A) , ϵ_{S} , ϵ_{A} ∈ {1 , −1} ,

where S and A are finite sets of symmetric or skew-symmetric tensors of order 2, respectively. (See Theorem 3.1.) To prove (1.13) we use the connection between tensors and the representation theory of symmetric groups. We discussed the formula (1.13) with P. B. Gilkey. In the course of this discussion P. B. Gilkey found another proof of (1.13) which is based on tensor algebra (see [26, pp. 41–44]). More precisely, P. B. Gilkey could show the stronger result that every algebraic curvature tensor has a representation

(1.14)  T = ∑_{S∈S} ϵ_{S} γ(S) , ϵ_{S} ∈ {1 , −1} ,

as well as a representation

(1.15)  T = ∑_{A∈A} ϵ_{A} α(A) , ϵ_{A} ∈ {1 , −1} .

In Section 5 we give a new proof of (1.14) and (1.15) using symmetry operators.

In Section 4 we verify our results by means of the Littlewood–Richardson rule and plethysms. We finish the paper with some remarks about applications of tensors γ(S) and α(A) in the theory of Osserman algebraic curvature tensors.

2. The symmetry class of algebraic curvature tensors

The set of the algebraic curvature tensors over V is a symmetry class in the sence of H. Boerner [4, p. 127]. We denote by 𝕂[S_{r}] the group ring of a symmetric group S_{r}. Every group ring element a = ∑_{p∈S_{r}} a(p) p ∈ 𝕂[S_{r}] acts as so-called symmetry operator on tensors T ∈ T_{r}V according to the definition

(2.1)  (aT)(v_{1}, …, v_{r}) := ∑_{p∈S_{r}} a(p) T (v_{p(1)}, …, v_{p(r)}) , v_{i} ∈ V .

Equation (2.1) is equivalent to

(2.2)  (aT)_{i_{1}...i_{r}} = ∑_{p∈S_{r}} a(p) T_{i_{p(1)}...i_{p(r)}} .

Definition 2.1. Let 𝔯 ⊆ 𝕂[S_{r}] be a right ideal of 𝕂[S_{r}] for which an a ∈ 𝔯 and a T ∈ T_{r}V exist such that aT ≠ 0. Then the tensor set

(2.3)  T_{𝔯} := {aT | a ∈ 𝔯 , T ∈ T_{r}V }

is called the symmetry class of tensors defined by 𝔯.

Since 𝕂[S_{r}] is semisimple for 𝕂 = ℝ, ℂ, every right ideal 𝔯 ⊆ 𝕂[S_{r}] possesses a generating idempotent e, i.e. 𝔯 fulfils 𝔯 = e · 𝕂[S_{r}].

Lemma 2.2. If e is a generating idempotent of 𝔯, then a tensor T ∈ T_{r}V belongs to T_{𝔯} iff

(2.4)  eT = T .

Thus we have

(2.5)  T_{𝔯} = {eT | T ∈ T_{r}V } .

Proof. Since every ideal element a ∈ 𝔯 can be written as a = e · x with x ∈ 𝕂[S_{r}], every aT ∈ T_{r}V fulfils aT = e(xT) = eT′ with T′ = xT ∈ T_{r}V. Thus we obtain T_{r} ⊆ {eT | T ∈ T_{r}V }. The relation {eT | T ∈ T_{r}V } ⊆ T_{r} is trivial. □

A Young frame of r is an arrangement of r boxes such that the numbers λ_{i} of boxes in the rows i = 1, …, l form a decreasing sequence λ_{1} ≥ λ_{2} ≥ … ≥ λ_{l} > 0 with λ_{1} + … + λ_{l} = r.

\begin{matrix}λ_{1} = 5 \\ λ_{2} = 4 \\ λ_{3} = 4 \\ λ_{4} = 2 \\ λ_{5} = 1 \end{matrix}

Obviously, a Young frame of r is characterized by a partition λ = (λ_{1}, …, λ_{l}) ⊢ r of r. A Young tableau t of a partition λ ⊢ r is the Young frame corresponding to λ which was fulfilled by the numbers 1, 2, …, r in any order.

t = 11 2 5 4 12 9 6 16 15 8 14 1 7 13 3 10

If a Young tableau t of a partition λ ⊢ r is given, then the Young symmetrizer y_{t} of t is defined by^{1}

(2.6)  y_{t} := ∑_{p∈H_{t}} ∑_{q∈V_{t}} sign(q) p ∘ q

where H_{t}, V_{t} are the groups of the horizontal or vertical permutations of t which only permute numbers within rows or columns of t, respectively. The Young symmetrizers of 𝕂[S_{r}] are essentially idempotent and define decompositions

(2.7) 𝕂[S_{r}] = ⨁_{λ⊢r} ⨁_{t∈ST_{λ}} 𝕂[S_{r}] · y_{t} , 𝕂[S_{r}] = ⨁_{λ⊢r} ⨁_{t∈ST_{λ}} y_{t} · 𝕂[S_{r}]

of 𝕂[S_{r}] into minimal left or right ideals. In (2.7), the symbol ST_{λ} denotes the set of all standard tableaux of the partition λ. Standard tableaux are Young tableaux in which the entries of every row and every column form an increasing number sequence.^{2}

S.A. Fulling, R.C. King, B.G.Wybourne and C.J. Cummins showed in [17] that the symmetry classes of the Riemannian curvature tensor R and its covariant derivatives ∇^{(u)}R are generated by special Young symmetrizers. To this end they assumed that the covariant derivatives ∇_{s_{1}}∇_{s_{2}} … ∇_{s_{u}}R_{ijkl} are symmetric in s_{1}, …, s_{u} for u ≥ 2. This is possible if one uses Ricci identities and neglect derivatives of orders smaller than u. Under this assumption one has

Theorem 2.3. Consider the Levi–Civita connection ∇ of a pseudo-Riemannian metric g. For u ≥ 0 the Riemann tensor and its covariant derivatives ∇^{(u)}R fulfil

(2.8)  e_{t}^{∗}∇^{(u)}R = ∇^{(u)}R

where e_{t} := y_{t}(u+1)/(2·(u+3)!) is an idempotent which is formed from the Young symmetrizer y_{t} of the standard tableau

(2.9)  t = \begin{matrix}1 & 3 & 5 & … & … & (u + 4) \\ 2 & 4 \end{matrix} .

in [34].

1We use the convention (p ∘ q)(i) := p(q(i)) for the product of two permutations p, q.

2About Young symmetrizers and Young tableaux see for instance [4, 5, 17, 19, 28, 29, 32, 33, 34, 35, 40, 41]. In particular, properties of Young symmetrizers in the case 𝕂 ≠ ℂ are described

The star ”∗” denotes the mapping

(2.10)  ∗ : a = ∑_{p∈S_{r}} a(p) p ↦ a^{∗} := ∑_{p∈S_{r}} a(p) p^{−1} .

A proof of this result of [17] can be found in [12], too. The proof needs only the symmetry properties (1.3), the identities Bianchi I and Bianchi II and Ricci identities. Thus Theorem 2.3 is a statement about the algebraic curvature tensors and ”algebraic” tensors that possess the symmetry properties of the covariant derivatives of the Riemann tensor. In particular, a tensor T ∈ T_{4}V is an algebraic curvature tensor iff T satisfies

(2.11)  y_{t}^{∗}T = 12 T

where y_{t} is the Young symmetrizer of the standard tableau

(2.12)  t = 1 3 2 4 .

3. A structure theorem for algebraic curvature tensors

Now we will show that every algebraic curvature tensor can be built from certain finite sets of symmetric and skew-symmetric tensors from T_{2}V.

Theorem 3.1. For every algebraic curvature tensor T ∈ T_{4}V there exist finite sets S and A of symmetric and alternating tensors S, A ∈ T_{2}V, respectively, such that

(3.1)  T = ∑_{S∈S} ϵ_{S} γ(S) + ∑_{A∈A} ϵ_{A} α(A) , ϵ_{S} , ϵ_{A} ∈ {1 , −1} .

Proof. Let T be an algebraic curvature tensor. Because of the symmetry T_{klij} = T_{ijkl}, every algebraic curvature tensor T fulfils fT = T, where f is the idempotent f := \frac{1}{2}(id + (1 3)(2 4)) ∈ 𝕂[S_{4}]. Thus every algebraic curvature tensor belongs to the symmetry class T_{𝔯′} of the right ideal 𝔯′ := f · 𝕂[S_{4}] and has a representation T = fT′ with T′ ∈ T_{4}V.

Since an arbitrary tensor T′ ∈ T_{4}V can be written as a finite sum of decomposable tensors a_{1} ⊗ a_{2} ⊗ a_{3} ⊗ a_{4} ∈ T_{4}V, we can use a representation

(3.2)  T′ = ∑_{(M,N)∈P} M ⊗ N

for a T′ ∈ T_{4}V, where P is a finite set of pairs (M, N) of tensors M, N ∈ T_{2}V. From (3.2) and T = fT′ we obtain

T = fT′ = \frac{1}{2} ∑_{(M,N)∈P}(M ⊗ N + N ⊗ M)

= \frac{1}{2} ∑_{(M,N)∈P} ((M + N) ⊗ (M + N) − M ⊗ M − N ⊗ N) .

Thus every tensor T ∈ T_{𝔯′} has a representation

(3.3)  T = \frac{1}{2} ∑_{M′∈M′} z_{M′} M′ ⊗ M′ , z_{M′} ∈ ℤ

with a certain finite set M′ of tensors from T_{2}V. If we carry out a transformation M′ ↦ M := \sqrt{|z_{M′}|/2} M′ in every summand of (3.3), we obtain

(3.4)  T = ∑_{M∈M} ϵ_{M} M ⊗ M , ϵ_{M} ∈ {1 , −1}

from (3.3).

Since every matrix M has a unique decomposition into a symmetric matrix S and a skew-symmetric matrix A, i.e. M = S + A, (3.4) leads to

(3.5)  T = ∑_{S+A∈M} ϵ_{S+A} (S ⊗ S + S ⊗ A + A ⊗ S + A ⊗ A) .

Now, let y_{t} be the Young symmetrizer of the standard tableau (2.12). If we apply (2.11) to the representation (3.5) of an algebraic curvature tensor T, we obtain

(3.6) T = \frac{1}{12} y_{t}^{∗}T = \frac{1}{12} ∑_{S+A∈M} ϵ_{S+A} y_{t}^{∗} (S ⊗ S + S ⊗ A + A ⊗ S + A ⊗ A) .

Then a calculation^{3} yields

\frac{1}{12} y_{t}^{∗}(S ⊗ S) = γ(S)  ,  \frac{1}{12} y_{t}^{∗}(A ⊗ S) = 0 ,
\frac{1}{12} y_{t}^{∗}(A ⊗ A) = α(A)  ,  \frac{1}{12} y_{t}^{∗}(S ⊗ A) = 0 ,

and if we set ϵ_{S} := ϵ_{S+A}, ϵ_{A} := ϵ_{S+A}, we obtain (3.1). □

Corollary 3.2. If S ⊆ T_{2}V and A ⊆ T_{2}V are finite sets of symmetric or alternating tensors, respectively, then

(3.7)  T := ∑_{S∈S} ϵ_{S} γ(S) + ∑_{A∈A} ϵ_{A} α(A) , ϵ_{S} , ϵ_{A} ∈ {1 , −1}

is an algebraic curvature tensor.

Proof. Every summand of (3.7) is an algebraic curvature tensor since γ(S) = \frac{1}{12} y_{t}^{∗}(S ⊗ S) and α(A) = \frac{1}{12} y_{t}^{∗}(A ⊗ A). □

3This calculation can be done by hand or by a cooperation of the Mathematica package PERMS [14] with one of the tensor packages Ricci [31] or MathTensor [8]. See [9].

4. Verification by the Littlewood–Richardson rule and plethysms

We can verify the results about y_{t}^{∗}(S ⊗ S), y_{t}^{∗}(S ⊗ A), y_{t}^{∗}(A ⊗ S) and y_{t}^{∗}(A ⊗ A) by means of the Littlewood–Richardson rule^{4} and plethysms^{5}. To this end we form group ring elements from tensors (see B. Fiedler [13, Sec.III.1] and [10, 11, 15, 16]).

Definition 4.1. Any tensor T ∈ T_{r}V and any r-tuple b := (v_{1}, …, v_{r}) ∈ V^{r} of vectors from V induce a function T_{b} : S_{r} → 𝕂 according to the rule

(4.1)  T_{b}(p) := T (v_{p(1)}, …, v_{p(r)}) , p ∈ S_{r} .

We identify this function with the group ring element ∑_{p∈S_{r}} T_{b}(p) p ∈ 𝕂[S_{r}], which we denote by T_{b}, too.

These T_{b} fulfil^{6}

Corollary 4.2. If T ∈ T_{r}V, b ∈ V^{r} and a ∈ 𝕂[S_{r}], then

(4.2)  (aT)_{b} = T_{b} · a^{∗} .

If a tensor T belongs to a certain symmetry class, then its T_{b} lie in a certain left ideal^{6}.

Proposition 4.3. Let e ∈ 𝕂[S_{r}] be an idempotent. Then a T ∈ T_{r}V fulfils the condition eT = T iff T_{b} ∈ 𝔩 := 𝕂[S_{r}] · e^{∗} for all b ∈ V^{r}, i.e. all T_{b} of T lie in the left ideal 𝔩 generated by e^{∗}.

Obviously, the S_{b} and A_{b} of symmetric/skew-symmetric tensors of order 2 lie in the left ideals

(4.3)  ∀ b ∈ V^{2} : S_{b} ∈ 𝔩_{1} := 𝕂[S_{2}] · e_{1} , e_{1} := \frac{1}{2} (id + (1 2))
(4.4)  ∀ b ∈ V^{2} : A_{b} ∈ 𝔩_{2} := 𝕂[S_{2}] · e_{2} , e_{2} := \frac{1}{2} (id − (1 2))

e_{1} and e_{2} are the (normalized) Young symmetrizers of the standard tableaux

t_{1} := 1 2  ,  t_{2} := 1 2

and the left ideals 𝔩_{1} and 𝔩_{2} are 1-dimensional, minimal left ideals, which belong to the equivalence classes of minimal left ideals characterized by the partitions λ_{1} = (2) and λ_{2} = (1^{2}), respectively. Then the group ring elements (A ⊗ S)_{b}, (S ⊗ A)_{b}, (S ⊗ S)_{b} and (A ⊗ A)_{b} belong to left ideals which are characterized by Littlewood–Richardson products [2][1^{2}] or plethysms [2] ⊙ [2], [1^{2}] ⊙ [2] (see B. Fiedler [13, Sec.III.3.2] or [15, Sec.4.2], [16]). Especially, we obtain the following structures for these ideals:

4See D. E. Littlewood [32, pp. 94–96], A. Kerber [29, Vol.240/p. 84], G. D. James and A. Kerber [28, p. 93], A. Kerber [30, Sec.5.5], I. G. Macdonald [33, Chap. I, Sec. 9], W. Fulton and J. Harris [18, pp. 455–456], S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins [17]. See also B. Fiedler [13, Sec.II.5].

5See A. Kerber [29], G. D. James and A. Kerber [28], A. Kerber [30], F. Sänger [39], D. E. Littlewood [32], I. G. Macdonald [33], S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins [17]. See also B. Fiedler [13, Sec.II.6].

6See B. Fiedler [13, Sec.III.3.1] and [10, 11, 15, 16].

\begin{matrix}(S ⊗ S)_{b} & [2] ⊙ [2] ∼ [4] + [2^{2}] & 𝔩^{(1)} = 𝔩_{[4]}^{(1)} ⊕ 𝔩_{[2^{2}]}^{(1)} \\ (S ⊗ A)_{b} & [2][1^{2}] ∼ [3 1] + [2 1^{2}] & 𝔩^{(2)} = 𝔩_{[3 1]}^{(2)} ⊕ 𝔩_{[2 1^{2}]}^{(2)} \\ (A ⊗ S)_{b} & [2][1^{2}] ∼ [3 1] + [2 1^{2}] & 𝔩^{(3)} = 𝔩_{[3 1]}^{(3)} ⊕ 𝔩_{[2 1^{2}]}^{(3)} \\ (A ⊗ A)_{b} & [1^{2}] ⊙ [2] ∼ [2^{2}] + [1^{4}] & 𝔩^{(4)} = 𝔩_{[2^{2}]}^{(4)} ⊕ 𝔩_{[1^{4}]}^{(4)} \end{matrix}

The relation [2][1^{2}] ∼ [3 1]+[2 1^{2}] can be determined by the Littlewood–Richardson rule. The relation [2] ⊙ [2] ∼ [4] + [2^{2}] follows from the formula

(4.5)  [2] ⊙ [n] ∼ ∑_{λ⊢n} [2 λ]

(see G.D. James and A. Kerber [28, p. 224]) and [1^{2}] ⊙ [2] ∼ [2^{2}] + [1^{4}] is a consequence of (4.5) and

(4.6) [μ] ⊙ [ν] ∼ ∑_{λ⊢m·n} m_{λ}[λ′] ⇔ [μ′] ⊙ [ν] ∼ ∑_{λ⊢m·n} m_{λ}[λ]
μ ⊢ m , ν ⊢ n , m, n, m_{λ} ∈ ℕ , m even .

(See F. Sänger [39, pp. 10–11]. See also B. Fiedler [13, Corr.II.6.8].) In (4.6) λ′ denotes the associated or transposed partition of a partition λ. All formulas of the above table can be determined by the Mathematica package Perms [14], too.

Every of the left ideals 𝔩^{(k)}, k = 1, 2, 3, 4, decomposes into two minimal left ideals 𝔩^{(k)} = 𝔩_{[μ_{k}]}^{(k)} ⊕ 𝔩_{[ν_{k}]}^{(k)}, where the partitions μ_{k} and ν_{k} charakterize the equivalence class

of minimal left ideals to which such an ideal 𝔩_{[μ_{k}]}^{(k)} , 𝔩_{[ν_{k}]}^{(k)} belongs.

Now the formula (4.2) yields that the group ring elements (y_{t}^{∗}(S ⊗ S))_{b}, (y_{t}^{∗}(A ⊗ S))_{b}, (y_{t}^{∗}(S ⊗ A))_{b} and (y_{t}^{∗}(A ⊗ A))_{b} are contained in one of the left ideals 𝔩^{(k)} · y_{t}. Since the Young symmetrizer y_{t} of (2.12) generates a minimal left ideal of the equivalence class of λ = (2^{2}) ⊢ 4, we obtain 𝔩_{[μ]}^{(k)} · y_{t} = {0} if μ ≠ (2^{2}). This leads to 𝔩^{(2)} · y_{t} = {0}, 𝔩^{(3)} · y_{t} = {0} and y_{t}^{∗}(A ⊗ S) = 0, y_{t}^{∗}(S ⊗ A). Furthermore, we obtain 𝔩_{[4]}^{(1)} · y_{t} = {0}, 𝔩_{[1^{4}]}^{(4)} · y_{t} = {0}. At most the products 𝔩_{[2^{2}]}^{(1)} · y_{t}, 𝔩_{[2^{2}]}^{(4)} · y_{t} are candidates for non-vanishing results. We have already seen in the proof of Theorem 3.1 that these cases really lead to non-vanishing expressions (1.6) and (1.7).

5. A new proof of the stronger result of P. B. Gilkey

Now we give a new proof of the following result of P. B. Gilkey (see [26, pp. 41–44]).

Theorem 5.1. For every algebraic curvature tensor T ∈ T_{4}V there exist finite sets S and A of symmetric and alternating tensors S, A ∈ T_{2}V, respectively, such that T has a representation

(5.1)  T = ∑_{S∈S} ϵ_{S} γ(S) , ϵ_{S} ∈ {1 , −1} ,

as well as a representation

(5.2)  T = ∑_{A∈A} ϵ_{A} α(A) , ϵ_{A} ∈ {1 , −1} .

Proof. We consider the following group ring elements

(5.3)  σ_{ϵ} := y_{t}^{∗} · f · ξ_{ϵ} , ϵ ∈ {1 , −1} ,

where y_{t} is the Young symmetrizer of the Young tableau (2.12) and

(5.4)  f := id + (1 3)(2 4)
(5.5)  ξ_{ϵ} := (id + ϵ (1 2)) · (id + ϵ (3 4)) , ϵ ∈ {1 , −1} .

A calculation (by means of PERMS [14]) shows that σ_{1} ≠ 0 and σ_{−1} ≠ 0 (see [9]).

The right ideals σ_{ϵ} · 𝕂[S_{4}] are non-vanishing subideals of the right ideal 𝔯 = y_{t}^{∗} · 𝕂[S_{4}] which defines the symmetry class T_{𝔯} of algebraic curvature tensors. Since 𝔯 is a minimal right ideal, we obtain 𝔯 = σ_{1} · 𝕂[S_{4}] = σ_{−1} · 𝕂[S_{4}], i.e. σ_{1} and σ_{−1} are generating elements of 𝔯, too.

A tensor T ∈ T_{4}V is an algebraic curvature tensor iff there exist a ∈ 𝔯 and T′ ∈ T_{4}V such that T = aT′. Since further every a ∈ 𝔯 has a representation a = σ_{ϵ} · x_{ϵ}, ϵ = 1, −1, a tensor T ∈ T_{4}V is an algebraic curvature tensor iff there is a tensor T′ ∈ T_{4}V such that T = σ_{1}T′ or T = σ_{−1}T′.

Let us consider the case ϵ = 1. We obtain all algebraic curvature tensors if we form T = σ_{1}T′, T′ ∈ T_{4}V. As in the proof of Theorem 3.1 we can use a representation

(5.6)  T′ = ∑_{(M,N)∈P} M ⊗ N

for a T′ ∈ T_{4}V, where P is a finite set of pairs (M, N) of tensors M, N ∈ T_{2}V. From (5.6) we obtain

(5.7)  ξ_{1}T′ = ∑_{(S′,S′′)∈P̃} S′ ⊗ S′′

where P̃ is the finite set of pairs (S′, S′′) of the symmetrized tensors S′ = M + M^{t}, S′′ = N + N^{t}, (M, N) ∈ P. Further the application of the symmetry operator f to ξ_{1}T′ yields

f(ξ_{1}T′) = ∑_{(S′,S′′)∈P̃}(S′ ⊗ S′′ + S′′ ⊗ S′)

= ∑_{(S′,S′′)∈P̃}((S′ + S′′) ⊗ (S′ + S′′) − S′ ⊗ S′ − S′′ ⊗ S′′)

= ∑_{S′′′∈S′′′} z_{S′′′} S′′′ ⊗ S′′′ , z_{S′′′} ∈ ℤ
(5.8)  = ∑_{S∈S} ϵ_{S} S ⊗ S , ϵ_{S} ∈ {1 , −1} .

The last step includes a transformation S′′′ ↦ S := \sqrt{|z_{S′′′}|}S′′′. The application of y_{t}^{∗} finishs the proof of (5.1):

T = σ_{1}T′ = (y_{t}^{∗} · f · ξ_{1})T′ = ∑_{S∈S} ϵ_{S} y_{t}^{∗}(S ⊗ S) = 12 ∑_{S∈S} ϵ_{S} γ(S) .

The proof of (5.2) can be carried out in exactly the same way. We have only to replace (5.7) by

(5.9)  ξ_{−1}T′ = ∑_{(A′,A′′)∈P̂} A′ ⊗ A′′

where P̂ is the finite set of pairs (A′, A′′) of the skew-symmetrized tensors A′ = M − M^{t}, A′′ = N − N^{t}, (M, N) ∈ P. □

Remark 5.2. The new generating elements σ_{1}, σ_{−1} of the right ideal 𝔯 do not automatically lead to new generating idempotents of 𝔯. A calculation [9] by means of PERMS [14] yields

y_{t}^{∗} · y_{t}^{∗} = 12 y_{t}^{∗}
y_{t}^{∗} · σ_{1} = 12 σ_{1}
σ_{1} · y_{t}^{∗} = 0
y_{t}^{∗} · σ_{−1} = 12 σ_{−1}
σ_{−1} · y_{t}^{∗} = 96 y_{t}^{∗}
σ_{1} · σ_{1} = 0
σ_{−1} · σ_{−1} = 96 σ_{−1}
σ_{−1} · σ_{1} = 96 σ_{1}
σ_{1} · σ_{−1} = 0 .

Thus σ_{−1} is an essencially idempotent element from which a generating idempotent of 𝔯 can be determined. However, σ_{1} is a nilpotent generating element of 𝔯.

6. Some applications of α and γ

We finish our paper with some remarks which arise from suggestions and hints of P. B. Gilkey.

In [26, pp. 191–193] P. B. Gilkey presents a family of algebraic curvature tensors which is a generalization of the family (1.11) in the case of a metric g with an arbitrary signature (p, q). This new family can be used to show that there are Jordan Osserman algebraic curvature tensors T for which J_{T} can have an arbitrary complicated Jordan normal form. We give a short view of the construction of Gilkey.

We consider a metric g ∈ T_{2}V of signature (p, q) on V.

Definition 6.1. An algebraic curvature tensor T ∈ T_{4}V is called spacelike Jordan Osserman (respectively timelike Jordan Osserman) if the Jordan normal form of the Jacobi operator J_{T} (x) is constant on S^{+}(V) := {x ∈ V | g(x, x) = +1} (respectively S^{−}(V) := {x ∈ V | g(x, x) = −1}). T is called Jordan Osserman if T is both spacelike Jordan Osserman and timelike Jordan Osserman.

Now we generalize the Clifford commutation relation (1.8)

Definition 6.2. Let V and W be vector spaces which possess metrics with signature (p, q) and (r, s), respectively, where r ≤ p and s ≤ q. We say that V admits a Clifford module structure Cliff(W) if there exist skew-symmetric linear transformations C_{i} : V → V for 1 ≤ i ≤ r + s so that

(1) C_{i}C_{j} + C_{j}C_{i} = 0 for i ≠ j.

(2) C_{i}^{2} = Id for r values of i.

(3) C_{i}^{2} = −Id for s values of i.

Note that a linear map C : V → V is called skew-symmetric if

∀ x, y ∈ V : g(Cx, y) = −g(x, Cy).

Now the following Lemma holds

Lemma 6.3. Let V and W possess metrics of signature (p, q) and (r, s), respectively (r ≤ p, s ≤ q). Assume that V admits a Cliff(W) module structure. Let J : W → W be a self-adjoint linear map of W. Then there exists a Jordan Osserman algebraic curvature tensor T on V so that the Jacobi operator J_{T} (x) is conjugate to g(x, x) J ⊕ 0 for every x ∈ S^{±}(V), where g is the metric of V.

In Gilkey’s proof of Lemma 6.3 the above Jordan Osserman algebraic curvature tensor T is given by the formula

(6.1)  T = ∑_{i} c_{i} α(A_{i}) + \frac{1}{2} ∑_{i≠j} c_{ij} α(A_{i} + A_{j})

where c_{i} and c_{ij} = c_{ji} are constants. The A_{i} are tensors of order 2 on V defined by A_{i}(x, y) := g(C_{i}x, y). Obviously the A_{i} are skew-symmetric.

The expression J ⊕ 0 is defined as follows: Under the assumtions of Lemma 6.3 we can find an isometry between V and W ⊕ ℝ^{(p−r,q−s)}. Then J ⊕ 0 is defined by setting

(J ⊕ 0)(w ⊕ z) := Jw ⊕ 0.

The sum (6.1) is a representation (5.2) of an algebraic curvature tensor which is formed by means of α from skew-symmetric tensors A_{i} ∈ T_{2}V.

Now the following Theorem by Gilkey tells us that we can construct Jordan Osserman algebraic curvature tensors with arbitrarily complicated Jordan normal form if we greatly increase the dimension by suitable powers of 2.

Theorem 6.4. Let J : W → W be an arbitrary linear map of a vector space W of dimension m. Then there exist l = l(m) and a Jordan Osserman algebraic curvature tensor T on V = ℝ^{(2^{l},2^{l})} so that J_{T} (x) is conjugate to ±J ⊕ 0 if x ∈ S^{±}(V).

If we choose a suitable J we can generate a Jacobi operator J_{T} (x) with an arbitrarily complicated Jordan normal form. In particular we can construct nondiagonalizable Jacobi operators. Thus the family (6.1) is different from the family (1.11) since the Jacobi operators of the tensors (1.11) are defined on a vector space V with a positive definite metric and can be diagonalized as self-adjoint operators on V.

In [26, p. 191] P. B. Gilkey presents also a second generalization of (1.11), (1.12) in the case of certain indefinite metrics. The algebraic curvature tensors of this generalization have a structure

(6.2)  T = λ_{0} γ(g) + ∑_{i} λ_{i} α(A_{i}) , λ_{0}, λ_{i} = const.

which is very similar to (1.12). The A_{i} ∈ T_{2}V are alternating tensors whose corresponding skew-symmetric maps C_{i} : V → V satisfy the Clifford commutation relations (1.8). The Jacobi operators of these tensors (6.2) are diagonalizable.

In a further hint P. B. Gilkey pointed out that there is a simple possibility to generate examples of Osserman algebraic curvature tensors of the form γ(S) or α(A) (S, A ∈ T_{2}V symmetric or alternating) if we have a pseudo-Riemannian metric g of indefinite signature (p, q). In particular, we can obtain in this way Osserman algebraic curvature tensors not of the form given in (1.11) or (6.2) (see Remark 6.7).

Proposition 6.5. Assume that V has a metric g with signature (p, q). (1) If p, q ≥ 1 and C : V → V is a symmetric map of V with C^{2} = 0, then

(6.3)  T := γ(S) , S(x, y) := g(Cx, y) ∀ x, y ∈ V

is an Osserman algebraic curvature tensor.

(2) If p, q ≥ 2 and C : V → V is a skew-symmetric map of V with C^{2} = 0, then

(6.4)  T := α(A) , A(x, y) := g(Cx, y) ∀ x, y ∈ V

is an Osserman algebraic curvature tensor.

(3) The Jacobi operators J_{T} (x) of (6.3) and (6.4) are nilpotent of order 2, i.e. J_{T} (x)^{2} = 0 for all x ∈ V.

Note that the tensors S, A ∈ T_{2}V defined by (6.3), (6.4), respectively, are symmetric or skew-symmetric, respectively.

To prove Proposition 6.5 we show first the existence of the above nilpotent symmetric or skew-symmetric maps C. Furhtermore we show that such maps C do not exist in the positive definite case.

Lemma 6.6. (1) If M is a symmetric or skew-symmetric m × m-matrix with

M^{2} = 0, then M = 0.

(2) Let p, q ≥ 2, m = p + q, and F be the m × m-diagonal matrix

(6.5)  F := diag(\underbrace{1, … 1}_{p}, \underbrace{−1, …, −1}_{q})

Then we can find non-vanishing symmetric or skew-symmetric m × mmatrices S, A, respectively, such that

(6.6)  (S · F)^{2} = 0 , (A · F)^{2} = 0 .

The assertion about a symmetric matrix S holds also for p, q ≥ 1.

Proof. Ad (1): If a matrix M = (m_{ij}) is symmetric or skew-symmetric, we have M^{T} = ϵ M, ϵ ∈ {1, −1}. Then M^{2} = 0 leads to ϵ M · M^{T} = 0 and

(6.7)  0 = trace M · M^{T} = ∑_{i,j} m_{ij}^{2} .

From (6.7) we obtain m_{ij} = 0 and M = 0.

Ad (2): The existence of skew-symmetric matrices A with (A · F)^{2} = 0 is guaranteed by an example of P. B. Gilkey [26, p. 186] which was constructed for p, q ≥ 2.

Let us now consider symmetric matrices under p, q ≥ 1. If we write F as block matrix

F = ( ... ; \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ; ... )

and choose a corresponding symmetric block matrix

S = ( 0 1 1 1 1 0 )

then we obtain

S · F = ( 0 ; \begin{matrix}1 & −1 \\ 1 & −1 \end{matrix} ; 0 ) .

Obviously, the matrix S · F satisfies (S · F)^{2} = 0. □

A relation B(x, y) = g(Cx, y) between a linear map C : V → V and a tensor B ∈ T_{2}V leads to relations

(6.8)  B_{ij} = C_{i}^{k} g_{kj} or C_{i}^{j} = B_{ik} g^{kj}

for the coordinates of B, C, and g. If we use an orthonormal basis of V then we have (g^{ij}) = diag(1, …, 1, −1, …, −1) and the matrix (B_{ik} g^{kj}) is a matrix product of the type B · F considered in Lemma 6.6. Thus statement (2) of Lemma 6.6 guarantees the existence of symmetric or skew-symmetric maps C with C^{2} = 0 in the above pseudo-Riemannian settings. Furthermore we see from statement (1) of Lemma 6.6 that such maps do not exist if g is positive definite.

Now we can prove Proposition 6.5.

Proof. For a symmetric map C : V → V the definitions (1.5), (1.6) and S(x, w) = g(Cx, w) lead to

g(J_{γ(S)}(x)y, w) = \frac{1}{3}(S(y, w)S(x, x) − S(x, y)S(x, w))

and

(6.9)  J_{γ(S)}(x)y = \frac{1}{3}(g(Cx, x)Cy − g(Cy, x)Cx) .

If now C^{2} = 0, then we obtain CJ_{γ(S)}(x)y = \frac{1}{3}(g(Cx, x)C^{2}y − g(Cy, x)C^{2}x) = 0 and J_{γ(S)}(x)^{2}y = \frac{1}{3}(g(Cx, x)CJ_{γ(S)}(x)y − g(CJ_{γ(S)}(x)y, x)Cx) = 0, i.e. J_{γ(S)}(x) is nilpotent of order 2 for all x ∈ V. But then all eigenvalues of J_{γ(S)}(x) are equal to zero, i.e. γ(S) is Osserman.

For a skew-symmetric map C : V → V we obtain from (1.5), (1.7), A(x, w) = g(Cx, w) and A(x, x) = 0

g(J_{α(A)}(x)y, w) = \frac{1}{3}(2A(y, x)A(x, w) + A(y, x)A(x, w) − A(y, w)A(x, x))
= A(y, x)A(x, w)

and

J_{α(A)}(x)y = g(Cy, x)Cx .

Again, the condition C^{2} = 0 leads to CJ_{α(A)}(x)y = 0 and J_{α(A)}(x)^{2}y = 0 for all x ∈ V. Consequently all eigenvalues of J_{α(A)}(x) vanish for all x ∈ V and α(A) is Osserman. □

Remark 6.7. If we use a representation of (1.11) or (6.2) in which we arrange the skew-symmetric maps C_{i} as in (1.9) then we will see that the Jacobi operator of (1.11) or (6.2) has the eigenvalues

0 , λ_{0} , λ_{0} − 3λ_{i} .

on the (pseudo)-unit sphere (see [24] and [26, p. 191]). The vanishing of all these eigenvalues leads to T = 0 for (1.11) and (6.2). Proposition 6.5, however, yields examples of non-vanishing Osserman algebraic curvature tensors γ(S), α(A) whose Jacobi operators have only the eigenvalues zero. Consequently the examples of Proposition 6.5 can not be transformed into a representation (1.11) or (6.2).

Moreover, the Jacobi operators of the algebraic curvature tensors (1.11) and (6.2) are diagonalizable (see [24] and [26, p. 191]). Since the Jacobi operators J_{γ(S)}, J_{α(A)} from Proposition 6.5 have only the eigenvalue zero, a Jacobi operator J_{γ(S)}, J_{α(A)} according to Proposition 6.5 is equal to zero if it is diagonalizable. Thus every algebraic curvature tensor γ(S), α(A) from Proposition 6.5, which has a non-vanishing Jacobi operator, is different from every algebraic curvature tensor with a diagonalizable Jacobi operator.

Remark 6.8. Osserman tensors (6.3) for metrics g with Lorentzian signature (1, q) are a special case. It is known that an Osserman algebraic curvature tensor has constant sectional curvature if the metric g has Lorentzian signature (1, q), q ≥ 1 (see [1, 20]). The Jacobi operator of an algebraic curvature tensor with constant sectional curvature is diagonalizable. Thus we obtain from Remark 6.7 that every algebraic curvature tensor γ(S) from Proposition 6.5 has a Jacobi operator J_{γ(S)} = 0 if p = 1 or q = 1. We verify this fact.

Lemma 6.9. Let F be the m × m-diagonal matrix

F := diag(1, −1, …, −1) .

(1) If S ≠ 0 is a symmetric m × m-matrix with (S · F)^{2} = 0 then there exists an invertible matrix D and a λ ≠ 0 such that

(6.10) D · S · D^{−1} = ( \begin{matrix}λ & ±λ \\ ±λ & λ \end{matrix} ; 0 ) and D · F · D^{−1} = F .

(2) A skew-symmetric m × m-matrix A with (A · F)^{2} = 0 vanishes.

Proof. Ad (1): We write a symmetric m × m-matrix S as block matrix

(6.11)  S = ( \begin{matrix}a & b \\ b^{T} & S̃ \end{matrix} ) ,

where S̃ is a symmetric (m − 1) × (m − 1)-matrix. Then we can transform S̃ into a diagonal matrix by a conjugation of S by a matrix

D = ( \begin{matrix}1 & 0 \\ 0^{T} & D̃ \end{matrix} )

where D̃ is a suitable orthogonal (m − 1) × (m − 1)-matrix. Obviously it holds D · F · D^{−1} = F for such a D.

From (6.11) we obtain

S · F = ( \begin{matrix}a & −b \\ b^{T} & −S̃ \end{matrix} ) , (S · F)^{2} = ( \begin{matrix}a^{2} − b · b^{T} & −a b + b · S̃ \\ a b^{T} − S̃ · b^{T} & −b^{T} · b + S̃^{2} \end{matrix} ) .

Assume that S̃ = 0. Then the condition (S · F)^{2} = 0 leads to the relations a^{2} −b·b^{T} = 0 and ab = 0 from which S = 0 follows. However this is a contradiction to S ≠ 0.

In case of S̃ ≠ 0 the matrix S̃ has rank 1 because of b^{T} · b − S̃^{2} = 0. Then S̃ has one and only one non-vanishing eigenvalue λ. Furthermore we obtain b ≠ 0 from b^{T} · b − S̃^{2} = 0.

Now the condition ab^{T} − S̃ · b^{T} = 0 means that b^{T} is an eigenvector of S̃ which belongs to the eigenvalue a. The value a = 0 is impossible since then a^{2} −b·b^{T} = 0 yields b = 0. Thus we obtain a = λ ≠ 0.

If we place λ ≠ 0 in the left upper corner of S̃, then b is an (m − 1)-tuple b = (μ, 0, …, 0) and a^{2} − b · b^{T} = 0 yields μ = ±λ.

Ad (2): We write a skew-symmetric m × m-matrix A as block matrix

(6.12)  A = ( \begin{matrix}0 & b \\ −b^{T} & Ã \end{matrix} ) ,

where Ã is a skew-symmetric (m − 1) × (m − 1)-matrix. From (6.12) we obtain

A · F = ( \begin{matrix}0 & −b \\ −b^{T} & −Ã \end{matrix} ) , (A · F)^{2} = ( \begin{matrix}b · b^{T} & b · Ã \\ Ã · b^{T} & b^{T} · b + Ã^{2} \end{matrix} ) .

Now the condition (A · F)^{2} = 0 yields b · b^{T} = 0, i.e. b = 0. But then Ã is a skew-symmetric matrix which has to fulfil Ã^{2} = 0. From Lemma 6.6 we obtain Ã = 0. □

Now let g be a Lorentzian metric with signature (1, q) and T = γ(S) be an Osserman algebraic curvature tensor according to Proposition 6.5. Then S can be transformed into a form (6.10) and the corresponding symmetric map

C := S · F = ( \begin{matrix}λ & ∓λ \\ ±λ & −λ \end{matrix} ; 0 )

has rank 1. Let us consider the Jacobi operator (6.9). If Cx = 0 then J_{γ(S)}(x) = 0. If Cx ≠ 0 then the vectors Cx and Cy are proportional since C has a 1-dimensional range. Again we obtain J_{γ(S)}(x) = 0. Thus J_{γ(S)}(x) ≡ 0 for every T = γ(S) according to Proposition 6.5 in the Lorentzian setting. Statement (2) of Lemma 6.9 tells us that a non-vanishing Osserman tensor T = α(A) of the type (6.4) does not exist in the case of a Lorentzian signature.

Acknowledgements. I would like to thank Prof. P. B. Gilkey for important and helpful discussions and for valuable suggestions for future investigations.

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Bernd Fiedler, Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany

URL: http://home.t-online.de/home/Bernd.Fiedler.RoschStr.Leipzig/
E-mail address: Bernd.Fiedler.RoschStr.Leipzig@t-online.de