Séminaire Lotharingien de Combinatoire 49 (2003), Article B49f

ON THE SYMMETRY CLASSES OF THE FIRST COVARIANT DERIVATIVES OF TENSOR FIELDS

BERND FIEDLER

Abstract. We show that the symmetry classes of torsion-free covariant derivatives ∇T of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products σ[1] where σ is a representation of the symmetric group S_{r} which is connected with the symmetry class of T. If σ ∼ [λ] is irreducible then σ[1] has a multiplicity free reduction [λ][1] ∼ ∑_{λ⊂μ}[μ] and all primitive idempotents belonging to that sum can be calculated from a generating idempotent e of the symmetry class of T by means of the irreducible characters or of a discrete Fourier transform of S_{r+1}. We apply these facts to derivatives ∇S, ∇A of symmetric or alternating tensor fields. The symmetry classes of the differences ∇S−sym(∇S) and ∇A−alt(∇A) = ∇A−dA are characterized by Young frames (r, 1) ⊢ r + 1 and (2, 1^{r−1}) ⊢ r + 1, respectively. However, while the symmetry class of ∇A − alt(∇A) can be generated by Young symmetrizers of (2, 1^{r−1}), no Young symmetrizer of (r, 1) generates the symmetry class of ∇S − sym(∇S). Furthermore we show in the case r = 2 that ∇S − sym(∇S) and ∇A − alt(∇A) can be applied in generator formulas of algebraic covariant derivative curvature tensors. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.

1. Introduction

The present paper continues our investigations of tensors by methods of Algebraic Combinatorics in [12, 13, 14]. The starting point are some questions concerning algebraic covariant derivative curvature tensors which arise from [13]. Many considerations which are necessary for an answer of these questions can be carried out for arbitrary tensor fields and lead to results about a connection of torsionfree covariant derivatives of differentiable tensor fields and Littlewood-Richardson products.

Algebraic curvature tensors are covariant tensors of order 4 which have the same algebraic properties as the Riemannian curvature tensor.

Definition 1.1. A covariant tensor ℜ of order 4 is called an algebraic curvature tensor iff its coordinates satisfy the conditions

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

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

A covariant tensor ℜ′ of order 5 is called an algebraic covariant derivative curvature tensor iff its coordinates fulfil

(1.3)  ℜ′_{ijklm} = −ℜ′_{jiklm} = −ℜ′_{ijlkm} = ℜ′_{klijm}
(1.4)  ℜ′_{ijklm} + ℜ′_{ikljm} + ℜ′_{iljkm} = 0
(1.5)  ℜ′_{ijklm} + ℜ′_{ijlmk} + ℜ′_{ijmkl} = 0 .

Relation (1.1) represents the index commutation symmetry of the Riemannian curvature tensor R whereas relations (1.2) and (1.5) correspond to the first and second Bianchi identity for the Riemann tensor

R_{ijkl} + R_{iklj} + R_{iljk} = 0
R_{ijkl; m} + R_{ijlm; k} + R_{ijmk; l} = 0 .

Investigations of algebraic curvature tensors were carried out by many authors. (See the extensive bibliography in the book [17] by P. B. Gilkey.) One of the problems which are considered in connection with algebraic curvature tensors is the search for generators of algebraic curvature tensors. In [12, 13, 14] we applied tools of Algebraic Combinatorics to treat this problem. In particular we used symmetry operators for tensors, given by elements a of the group ring 𝕂[S_{r}] of a symmetric group S_{r}, Young tableaux t, Young symmetrizers y_{t} and symmetry classes of tensors, defined by right ideals 𝔯 of 𝕂[S_{r}]. Details about these concepts can be found in Sections 2 and 4 and in [12, 13, 14].

P. B. Gilkey [17, pp.41-44] and B. Fiedler [12] gave different proofs that the vector space of all algebraic curvature tensors is spanned by each of the following types of tensors

(1.6)  y_{t}^{∗}(S ⊗ S) , y_{t}^{∗}(A ⊗ A)

which are defined by symmetric or alternating covariant tensors S or A of order 2. The vector space of algebraic covariant derivative curvature tensors is generated by each of the following tensor types

y_{t′}^{∗}(S ⊗ S′) , y_{t′}^{∗}(S′ ⊗ S)
(1.7)  y_{t′}^{∗}(U ⊗ S) , y_{t′}^{∗}(S ⊗ U)
y_{t′}^{∗}(U ⊗ A) , y_{t′}^{∗}(A ⊗ U)

(see^{1} B. Fiedler [13]). Here S, A are again symmetric or alternating tensors of order 2, S′ is a symmetric tensor of order 3 and U is a covariant tensor of order 3 from an irreducible symmetry class that belongs to the partition (2, 1) ⊢ 3 and is defined by a minimal right ideal different from the right ideal f · 𝕂[S_{3}] with the generating idempotent

(1.8)  f := \{\frac{1}{2} (id − (1 3)) − \frac{1}{6} y\} , y := ∑_{p∈S_{3}} sign(p) p .

1A first proof that y_{t′}^{∗}(S ⊗ S′) and y_{t′}^{∗}(S′ ⊗ S) are generators for ℜ′ was given by P. B. Gilkey [17, p.236].

In (1.6) and (1.7) y_{t} and y_{t′} denote the Young symmetrizers of the Young tableaux

(1.9)  t = 1 3 2 4  , t′ = 1 3 5 2 4 .

The generators (1.7) lead to the question whether there exist typical tensorial quantities of differential geometry which possess a symmetry of the same type as the above tensors U. It can be shown (see Section 3) that the differences

(1.10)  ∇S − sym(∇S) , ∇A − alt(∇A) = ∇A − dA

between the covariant derivatives ∇S, ∇A of symmetric/alternating covariant tensor fields S, A of order 2 and their symmetrized/anti-symmetrized covariant derivatives sym(∇S), alt(∇A) have such a symmetry^{2}. Furthermore computer calculations by means of the Mathematica packages Ricci [21] and PERMS [9] showed that the symmetry class of ∇A − alt(∇A) is generated by certain Young symmetrizers, for instance by the Young symmetrizer of the standard tableau

1 3 2 .

However, the same computation yields the surprising result that no Young symmetrizer of a Young frame (2 1) ⊢ 3 generates the symmetry class of ∇S − sym(∇S). One goal of the present paper is to find out whether the covariant derivatives of symmetric or alternating tensor fields of order r > 2 have such a behaviour, too.

In Section 2 we collect some basic facts about the connection between symmetry classes of covariant tensors of order r and left or right ideals of the group ring 𝕂[S_{r}] of the symmetric group S_{r}. In Section 3 we show that the symmetry class of a torsion-free covariant derivative ∇T of a differentiable tensor field T of order r is defined by a left ideal 𝔩 ⊆ 𝕂[S_{r+1}] which is the representation space of a Littlewood-Richardson product σ[1], where σ is a representation of S_{r} connected with the symmetry class of T. If σ ∼ [λ], λ ⊢ r, is irreducible then the Littlewood-Richardson rule yields a multiplicity-free decomposition

(1.11)  [λ][1] ∼ ∑_{{μ⊢r+1 \atop λ⊂μ}} [μ] .

In particular the symmetry classes of the covariant derivatives ∇S, ∇A of symmetric/alternating tensor fields S, A of order r are characterized by Littlewood-Richardson products

[r][1] ∼ [r + 1] + [r, 1] , [1^{r}][1] ∼ [1^{r+1}] + [2, 1^{r−1}] .

If we know a primitive generating idempotent e ∈ 𝕂[S_{r}] of the symmetry class of T, then all unique primitive idempotents h_{μ} ∈ 𝕂[S_{r+1}] corresponding to (1.11) can be calculated from e by means of the symmetrizers of the irreducible characters of S_{r+1} or, more efficiently, by a discrete Fourier transform.

2Note that alt(∇A) is equal to the exterior derivative dA of the alternating tensor field A.

In Section 4 we investigate the parts [r, 1], [2, 1^{r−1}] of ∇S, ∇A for arbitrary order r ≥ 2. We show for ∇S that no Young symmetrizer with a Young frame (r, 1) ⊢ r + 1 is a generator of the [r, 1]-part of ∇S. The [2, 1^{r−1}]-part of ∇A, however, is generated by the Young symmetrizer of the lexicographically greatest standard tableau of (2, 1^{r−1}) ⊢ r + 1 (see (4.2)) and every other standard tableau of (2, 1^{r−1}) ⊢ r + 1 annihilates the [2, 1^{r−1}]-part. Furthermore that [2, 1^{r−1}]-part is generated or annihilated by many other Young symmetrizers of non-standard tableaux of (2, 1^{r−1}) ⊢ r +1. We present complete computer generated lists of such Young symmetrizers for r = 2, 3, 4 in an Appendix.

The last Section of the paper deals with the question whether tensors (1.10) can be used as generators U of algebraic covariant derivative curvature tensors in formulas (1.7). Both S and A satisfy the condition that the symmetry classes of the tensors (1.10) are not generated by the above right ideal f · 𝕂[S_{3}] with generating idempotent (1.8). Thus all tensors (1.10) can play the role of U in (1.7).

2. Symmetry classes of tensors

Let 𝕂 be the field of real or complex numbers ℝ, ℂ. We denote by 𝕂[S_{r}] the group ring of a symmetric group S_{r}. Furthermore we consider the 𝕂-vector space T_{r}V of r-times covariant 𝕂-valued tensors T over a finite dimensional 𝕂-vector space V. 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. If 𝔯 is a minimal right ideal, then T_{𝔯} is called irreducible.

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. ^{3} If e is a generating idempotent of 𝔯, then a tensor T ∈ T_{r}V belongs to T_{𝔯} iff

(2.4)  eT = T .

3See H. Boerner [1, p.127] or B. Fiedler [11], [8, p.110].

Thus we have

(2.5)  T_{𝔯} = {eT | T ∈ T_{r}V } .

Symmetry classes can be characterized by left ideals of 𝕂[S_{r}], too. To see this, we construct group ring elements from tensors.

Definition 2.3. Every tensor T ∈ T_{r}V and every r-tuple b = (v_{1}, …, v_{r}) ∈ V^{r} of vectors from V induce a group ring element

(2.6)  T_{b} := ∑_{p∈S_{r}} T (v_{p(1)}, …, v_{p(r)}) p ∈ 𝕂[Sr] .

A connection between (2.1) and (2.6) is given by the formula^{4}

(2.7)  ∀ T ∈ T_{r}V , ∀ a ∈ 𝕂[S_{r}] , ∀ b ∈ V^{r} : (aT)_{b} = T_{b} · a^{∗}

where the star ’∗’ denotes the mapping

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

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

Proposition 2.4. ^{5} 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^{∗}. Moreover, if dim V ≥ r then 𝔩 is spanned by all group ring elements T_{b} of tensors T ∈ T_{𝔩^{∗}}, i.e. 𝔩 = L{T_{b} | T ∈ T_{𝔩^{∗}} , b ∈ V^{r}} where L denotes the linear closure.

Thus we can use the left ideal 𝔩 = 𝔯^{∗} instead of the right ideal 𝔯 to characterize the symmetry class of 𝔯.

3. Symmetry classes of the first covariant derivatives of tensor fields

Now we determine results about the symmetry classes of the first covariant derivatives of tensor fields. In particular, we are interested in symmetric or alternating covariant tensor fields.

We consider only differentiable objects of class C^{∞}. Let M be an m-dimensional differentiable manifold equipped with a linear connection or covariant derivative ∇. We denote by T_{r}M, r ≥ 0, the set of differentiable covariant tensor fields of order r on M. If T ∈ T_{r}M, r ≥ 1, then its covariant derivative ∇T has a coordinate representation^{6}

4See B. Fiedler [11] or [8, p.110].

5See B. Fiedler [8, Sec.III.3.1] and [5, 6, 10].

(3.1)  ∇_{i_{r+1}}T_{i_{1}...i_{r}} = ∂_{i_{r+1}}T_{i_{1}...i_{r}} − ∑_{k=1}^{r} Γ_{i_{r+1}i_{k}}^{s_{k}} T_{i_{1}...i_{k−1}s_{k}i_{k+1}...i_{r}}

where ∂ is a partial derivative of the coordinates of T and Γ_{ij}^{k} are the connection coefficients of ∇. Instead of (3.1) we write also

(3.2)  T_{i_{1}...i_{r} ; i_{r+1}} = T_{i_{1}...i_{r} , i_{r+1}} − ∑_{k=1}^{r} Γ_{i_{r+1}i_{k}}^{s_{k}} T_{i_{1}...i_{k−1}s_{k}i_{k+1}...i_{r}} .

Every tensor in a fixed point p of M can be gained as covariant derivative of a suitable tensor field.

Lemma 3.1. Let M_{p} be the tangent space of M in a point p ∈ M of M and W ∈ T_{r+1}M_{p}, r ≥ 0, be a covariant tensor of order r + 1 over M_{p}. Then we can find a covariant tensor field T ∈ T_{r}M such that (∇T)|_{p} = W.

Proof. First we consider the case r ≥ 1. In a suitable open neighbourhood U of p we can choose a chart x such that x(p) = 0. If W_{i_{1}...i_{r+1}} are the coordinates of W with respect to x and x^{i} are the coordinate functions of x then T̃_{i_{1}...i_{r}} := W_{i_{1}...i_{r}k}x^{k} yields a differentiable tensor field on U. Further we can consider a function ϕ of class C^{∞} on U for which open neighbourhoods U_{1} and U_{2} of p exist such that p ∈ U_{1} ⊂ U_{2} ⊂ U and ϕ|_{U_{1}} ≡ 1, ϕ|_{U\U_{2}} ≡ 0. By means of ϕ we obtain a differentiable tensor field T ∈ T_{r}M if we set T |_{U} := ϕT̃ and T |_{M\U} := 0. But T fulfils (∇T)|_{p} = W since we can write

T_{i_{1}...i_{r} ; i_{r+1}}(p) = T_{i_{1}...i_{r} , i_{r+1}}(p) − ∑_{k=1}^{r} Γ_{i_{r+1}i_{k}}^{s_{k}}(p) T_{i_{1}...i_{k−1}s_{k}i_{k+1}...i_{r}}(p) = W_{i_{i}...i_{r}i_{r+1}} .

The last equality follows from T_{i_{1}...i_{r}}(p) = 0 and T_{i_{1}...i_{r} , i_{r+1}}(p) = W_{i_{i}...i_{r}i_{r+1}}.

In the case r = 0 the tensor W has order 1 and T̃ = W_{k}x^{k} is a tensor field of order 0, i.e. a differentiable function. Obviously we can form the tensor field (function) T ∈ T_{0}M in the same way as in the case r ≥ 1 and we obtain T_{;i} = T_{,i} = W_{i} on the neighbourhood U_{1} of p. □

Lemma 3.1 leads to the consequence that every symmetry class can be generated by covariant derivatives of suitable tensor fields.

Corollary 3.2. Let 𝔯 ⊆ 𝕂[S_{r+1}], r ≥ 0, be a right ideal with generating idempotent e ∈ 𝕂[S_{r+1}] for which an a ∈ 𝔯 and a W ∈ T_{r+1}M_{p} exist such that aW ≠ 0. Then the symmetry class T_{𝔯} of tensors of order r + 1 over M_{p} fulfils

(3.3)  T_{𝔯} = {e(∇T)|_{p} | T ∈ T_{r}M} .

6We use the Einstein summation convention, i.e. a symbol such as T_{i_{1}...s... ...i_{r}}^{s} means T_{i_{1}...s... ...i_{r}}^{s} := ∑_{s=1}^{m} T_{i_{1}...s... ...i_{r}}^{s}.

Proof. According to (2.5) we have T_{𝔯} = {eW | W ∈ T_{r+1}M_{p}}. But for every W ∈ T_{r+1}M_{p} there exists a T ∈ T_{r}M such that W = (∇T)|_{p}. Thus Corollary 3.2 follows. □

The symmetrization ’_{(...)}’ and anti-symmetrization ’_{[...]}’ of a tensor field T of order r is defined by

(3.4)  T_{(i_{1}...i_{r})} := \frac{1}{r!} ∑_{p∈S_{r}} T_{i_{p(1)}...i_{p(r)}}

(3.5)  T_{[i_{1}...i_{r}]} := \frac{1}{r!} ∑_{p∈S_{r}} sign(p) T_{i_{p(1)}...i_{p(r)}} .

From now on we consider only covariant derivatives ∇ which are torsion-free, i.e. Γ_{[ij]}^{k} = 0. It is well-known for such ∇ that the operations ’_{(...)}’ or ’_{[...]}’ and the operator ∇ are permutable. However, this statement is correct for arbitrary symmetry operators a ∈ 𝕂[S_{r}], too.

Lemma 3.3. Let ∇ be torsion-free and a = ∑_{p∈S_{r}} a(p) p ∈ 𝕂[S_{r}], r ≥ 2. If we consider a T ∈ T_{r}M and set H_{i_{1}...i_{r}} := (aT)_{i_{1}...i_{r}} and W_{i_{1}...i_{r}i_{r+1}} := T_{i_{1}...i_{r} ; i_{r+1}} with respect to arbitrary local coordinates, then it holds

(3.6)  ∑_{p∈S_{r}} a(p) W_{i_{p(1)}...i_{p(r)} i_{r+1}} = H_{i_{1}...i_{r} ; i_{r+1}} .

Proof. Let q ∈ M be an arbitrary point of M. We can choose such coordinates arround q that all Γ_{ij}^{k} vanish in q, i.e. Γ_{ij}^{k}(q) = 0. Then we have W_{i_{1}...i_{r}i_{r+1}}(q) = T_{i_{1}...i_{r} , i_{r+1}}(q) and H_{i_{1}...i_{r} ; i_{r+1}}(q) = H_{i_{1}...i_{r} , i_{r+1}}(q). But since obviously

∑_{p∈S_{r}} a(p) T_{i_{p(1)}...i_{p(r)} , i_{r+1}}(q) = H_{i_{1}...i_{r} , i_{r+1}}(q) ,

we obtain Lemma 3.3. □

The version of Lemma 3.3 for the symmetry operators ’_{(...)}’ and ’_{[...]}’ reads

Lemma 3.4. Let ∇ be torsion-free and r ≥ 2. If we set S_{i_{1}...i_{r}} := T_{(i_{1}...i_{r})}, A_{i_{1}...i_{r}} := T_{[i_{1}...i_{r}]} and W_{i_{1}...i_{r}i_{r+1}} := T_{i_{1}...i_{r} ; i_{r+1}}, then it holds

(3.7)  W_{(i_{1}...i_{r}) i_{r+1}} = S_{i_{1}...i_{r} ; i_{r+1}}
(3.8)  W_{[i_{1}...i_{r}] i_{r+1}} = A_{i_{1}...i_{r} ; i_{r+1}} .

Proposition 3.5. Consider the case r ≥ 2. Let S̃_{r} := {p ∈ S_{r+1} | p(r +1) = r +1} be the subgroup of those permutations of S_{r+1} which have r +1 as fixed point. Then

(3.9)  e_{s} := \frac{1}{r!} ∑_{p∈S̃_{r}} p , e_{a} := \frac{1}{r!} ∑_{p∈S̃_{r}} sign(p) p

are idempotents of the group ring 𝕂[S_{r+1}] which fulfil

(3.10)  e_{s}^{∗} = e_{s} , e_{a}^{∗} = e_{a} .

If ∇ is a torsion-free covariant derivative and S, A are symmetric or alternating differentiable tensor fields of order r, respectively, then it holds^{7}

(3.11)  e_{s}^{∗}∇S = ∇S , e_{a}^{∗}∇A = ∇A .

Proof. Taking into account Lemma 3.4 and sign(p·q) = sign(p)sign(q), sign(p^{−1}) = sign(p), we can prove Proposition 3.5 by simple calculations. □

Now we will show that the symmetry classes of covariant derivatives of tensor fields are characterized by Littlewood-Richardson products. Let us denote by ω̆ the regular representation of the symmetric group S_{r+1}:

(3.12) ω̆ : S_{r+1} → Gl(𝕂[S_{r+1}]) , ω̆_{p}(a) = p · a , p ∈ S_{r+1} , a ∈ 𝕂[S_{r+1}] .

The left ideals 𝔩 ⊆ 𝕂[S_{r+1}] of 𝕂[S_{r+1}] can be considered representation spaces of subrepresentations ρ = ω̆|_{𝔩} of ω̆:

(3.13)  ρ : S_{r+1} → Gl(𝔩) , ρ_{p}(a) = p · a , p ∈ S_{r+1} , a ∈ 𝔩 .

We see from a generalization of Proposition 3.5 that investigations of covariant derivatives of tensor fields can be based on following

Setting 3.6. Let T ∈ T_{r}M be a differentiable tensor field of order r on M that lies everywhere in a symmetry class defined by a left ideal 𝕂[S_{r}] · e with generating idempotent e ∈ 𝕂[S_{r}], i.e. e^{∗}T = T on every tangent space M_{p} of M. We identify S_{r} with S̃_{r} by means of [i_{1}, …, i_{r}] ↦ [i_{1}, …, i_{r}, r + 1] (where [i_{1}, …, i_{r}] is the list representation of a permutation) and denote by ẽ ∈ 𝕂[S̃_{r}] the corresponding embedding of e ∈ 𝕂[S_{r}] into 𝕂[S_{r+1}]. If ∇ is a torsion-free covariant derivative on M then the left ideal 𝔩 := 𝕂[S_{r+1}] · ẽ defines symmetry classes in the T_{r+1}M_{p} that contain ∇T in every point of M, i.e. ẽ^{∗}∇T = ∇T on every tangent space M_{p}. Furthermore, we can consider the left ideal 𝔩̃:= 𝕂[S̃_{r}] · ẽ of 𝕂[S̃_{r}]. The left ideals 𝔩 and 𝔩̃ are the representation spaces of the representations^{8}

(3.14)  𝔩 ⟺ ρ := ω̆|_{𝔩}
(3.15)  𝔩̃ ⟺ σ := (ω̆ ↓ S̃_{r})|𝔩̃.

A simple consequence of Corollary 3.2, Lemma 3.3 and Setting 3.6 is that the symmetry class of the above ∇T is generated by the covariant derivatives of the symmetizations e^{∗}W of arbitrary tensor fields W ∈ T_{r}M.

7Here we assume that the tensor indices are numbered in the manner of (3.2).

8If α is a representation of a group G and H ⊆ G is a subgroup of G, then α ↓ H denotes the restriction of α to H.

Proposition 3.7. Assume that Setting 3.6 and r ≥ 1 are valid and p ∈ M. Then we have

T_{𝔩^{∗}} = {ẽ^{∗}(∇W)|_{p} | W ∈ T_{r}M} = {(∇e^{∗}W)|_{p} | W ∈ T_{r}M} .

Now we determine Littlewood-Richardson products describing covariant derivatives.

Theorem 3.8. Assume that Setting 3.6 and r ≥ 1 are valid. Then the representation ρ is a Littlewood-Richardson product

(3.16)  ρ ∼ σ [1] .

If σ is an irreducible representation σ ∼ [λ], λ ⊢ r, then the Littlewood-Richardson rule^{9} yields a multiplicity-free decomposition

(3.17)  ρ ∼ [λ][1] ∼ ∑_{{μ⊢r+1 \atop λ⊂μ}} [μ] ,

i.e. [λ][1] decomposes into a sum of all Young frames [μ] which can be formed from [λ] by adding one box to [λ].

Proof. When we introduce the notation S̃_{1} := {id} for the trivial subgroup of S_{r+1}, then the set product S̃_{r} = S̃_{r} · S̃_{1} is a direct product S̃_{r} = S̃_{r} × S̃_{1}. (It is even a Young subgroup.) We consider the representations

(3.18)  ι : S̃_{1} → Gl(𝕂[S̃_{1}]) , ι_{id}(u) = u , u ∈ 𝕂[S̃_{1}]
(3.19) σ : S̃_{r} → Gl(𝕂[S̃_{r}] · ẽ) , σ_{p}(v) = p · v , p ∈ S̃_{r} , v ∈ 𝕂[S̃_{r}] · ẽ.

Obviously we can regard σ as an outer tensor product of representations

σ = σ # ι .

Thus the left ideal 𝔩 = 𝕂[S_{r+1}]·ẽ is the representation space of the induced representation (σ # ι) ↑ S_{r+1} which has the structure of a Littlewood-Richardson product, i.e.

ρ = (σ # ι) ↑ S_{r+1} ∼ σ[1] .

If σ is irreducible, i.e. σ ∼ [λ], λ ⊢ r, then the Littlewood-Richardson rule yields (3.17) (see also Figure 1). □

Remark 3.9. If we restrict us to irreducible representations σ, then the proof of Theorem 3.8 is a repetition of a part of the proof of the branching theorem for irreducible representations of symmetric groups (see A. Kerber [19, Vol.240/p.85]).

9See D. E. Littlewood [22, pp.94-96], A. Kerber [19, Vol.240/p.84], G. D. James and A. Kerber [18, p.93], A. Kerber [20, Sec.5.5], I. G. Macdonald [23, Chap.I,Sec.9], R. Merris [24, p.100], W.

Fulton and J. Harris [16, pp.455-456], S. A. Fulling, R. C. King, B. G. Wybourne and C. J.

Cummins [15]. See also B. Fiedler [8, Sec.II.5].

Remark 3.10. If σ is a reducible representation and we know a decomposition σ = ⨁_{i} σ_{i} into subrepresentations σ_{i} (irreducible or reducible), then we can use the formula

(3.20)  σ[1] ∼ ∑_{i} σ_{i}[1]

to determine the structure of the decomposition of σ[1] into irreducible subrepresentations.

Remark 3.11. Additional information about a tensor field considered can lead to a further reduction of the sum (3.17). For instance it is well-known that the symmetry classes of the Riemannian curvature tensor R and its covariant derivative^{10} ∇R are defined by the Young symmetrizers^{11} y_{t} and y_{t′} of the Young tableaux^{12}

(3.21)  t = 1 3 2 4  , t′ = 1 3 5 2 4 .

However, if we apply (3.17) to the tableau t we obtain [2^{2}][1] ∼ [3, 2]+[2^{2}, 1] ≁ [3, 2]. The difference results from the fact that ∇R fulfils the second Bianchi identity

R_{ijkl; m} + R_{ijlm; k} + R_{ijmk; l} = 0

which is not satisfied by other tensor fields from T_{4}M in general.

A second example which shows such effects is the case of higher covariant derivatives of tensor fields. If we apply Theorem 3.8 to covariant derivatives^{13} of second order T_{i_{1}...i_{r} ; i_{r+1}i_{r+2}} of a tensor field T ∈ T_{r}M then Theorem 3.8 yields a result in which the so-called Ricci identity

(3.22)  T_{i_{1}...i_{r} ; [i_{r+1}i_{r+2}]} = \frac{1}{2} ∑_{k=1}^{r} R_{i_{r+1}i_{r+2}i_{k}}^{s_{k}} T_{i_{1}...i_{k−1}s_{k}i_{k+1}...i_{r}}

was left out of account. Thus the set of Young frames determined by multiple application of (3.17) will be ”too large”. (3.17) produces a set of Young frames which is correct also for covariant derivatives T_{i_{1}...i_{r}i_{r+1} ; i_{r+2}} of tensor fields T ∈ T_{r+1}M of order r + 1 to which an identity (3.22) is irrelevant.

Now we present a version of Theorem 3.8 for the special case of symmetric or alternating tensor fields.

Theorem 3.12. Assume that r ≥ 2. Let 𝔩_{s} := 𝕂[S_{r+1}] · e_{s}, 𝔩_{a} := 𝕂[S_{r+1}] · e_{a} be the left ideals generated by the idempotents e_{s}, e_{a}. Then the subrepresentations ρ_{s} := ω̆|_{𝔩_{s}}, ρ_{a} := ω̆|_{𝔩_{a}} are Littlewood-Richardson products ρ_{s} ∼ [r][1], ρ_{a} ∼ [1^{r}][1] for which the Littlewood-Richardson rule yields decompositions

(3.23)  [r][1] ∼ [r + 1] + [r, 1] , [1^{r}][1] ∼ [1^{r+1}] + [2, 1^{r−1}] .

10Here we assume that ∇ is the Levi-Civita connection of a pseudi-Riemannian fundamental tensor g ∈ T_{2}M and R is the curvature tensor of ∇.

11See Section 4 for some details about Young symmetrizers.

12See S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins [15]. See also B. Fiedler [7].

13Again we assume that ∇ is a Levi-Civita connection.

× ≡ ⇒  + ≡  ≡ × ≡ ⇒  + ≡  ≡

Figure 1. The Littlewood-Richardson products [r][1] and [1^{r}][1].

The idempotents e_{s}, e_{a} have unique decompositions corresponding to (3.23) into primitive orthogonal idempotents

(3.24)  e_{s} = f_{s} + h_{s} , e_{a} = f_{a} + h_{a}

which generate the minimal left ideals that are the representation spaces of the irreducible representations in (3.23):

𝕂[S_{r+1}] · f_{s} ⟺ [r + 1]
𝕂[S_{r+1}] · h_{s} ⟺ [r, 1]
𝕂[S_{r+1}] · f_{a} ⟺ [1^{r+1}]
𝕂[S_{r+1}] · h_{a} ⟺ [2, 1^{r−1}] .

We know all idempotents in (3.24) since

(3.25) f_{s} = \frac{1}{(r + 1)!} ∑_{p∈S_{r+1}} p , f_{a} = \frac{1}{(r + 1)!} ∑_{p∈S_{r+1}} sign(p) p .

Proof. If we apply Theorem 3.8 to the representations

(3.26) σ_{s} : S̃_{r} → Gl(𝕂[S̃_{r}] · e_{s}) , (σ_{s})_{p}(v) = p · v , p ∈ S̃_{r} , v ∈ 𝕂[S̃_{r}] · e_{s}
(3.27) σ_{a} : S̃_{r} → Gl(𝕂[S̃_{r}] · e_{a}) , (σ_{a})_{p}(v) = p · v , p ∈ S̃_{r} , v ∈ 𝕂[S̃_{r}] · e_{a}

then we obtain (3.23) since σ_{s} and σ_{a} are irreducible and fulfil σ_{s} ∼ [r], σ_{a} ∼ [1^{r}]. (See Figure 1 for the use of the Littlewood-Richardson rule.)

The relations (3.23) tell us that every of the representations (σ_{s} # ι) ↑ S_{r+1}, (σ_{a} # ι) ↑ S_{r+1} decomposes into two irreducible representations which have multiplicities 1. Thus the generating idempotents e_{s}, e_{a} of the representation spaces possess unique corresponding decompositions (3.24) into primitive othogonal idempotents. Since we know that the idempotents for representations [r + 1] and [1^{r+1}] are given by (3.25) we can calculate the remaining idempotents h_{s} and h_{a}, too. □

Remark 3.13. In the case of an alternating tensor field A ∈ T_{r}M the symmetry operator f_{a} transforms ∇A into the exterior derivative dA of A, i.e. f_{a}^{∗}(∇A) = f_{a}(∇A) = dA. Thus the symmetry operator h_{a} yields the difference of ∇A and dA, i.e. h_{a}^{∗}(∇A) = ∇A − dA.

Now let us consider the more general case of an arbitrary tensor field T ∈ T_{r}M whose symmetry class is defined by a known primitive idempotent e ∈ 𝕂[S_{r}]. Also in this case, there is a simple possibility to calculate all primitive idempotents which belong to a decomposition (3.17) for the covariant derivatives of T. A starting point is the well-known

Lemma 3.14. ^{14} Let λ ⊢ r be a partition of r ≥ 1 and χ_{λ} be the irreducible character of S_{r} which belongs to that equivalence class of irreducible representations of S_{r} which contains the irreducible representations^{15} ω̆|_{𝕂[S_{r}]·y_{t}} defined by Young symmetrizers y_{t} with Young frame λ. Then the group ring element

(3.28)  e_{λ} := \frac{χ_{λ}(id)}{r!} ∑_{p∈S_{r}} χ_{λ}(p) p^{−1}

is the unique centrally primitive idempotent that generates the minimal two-sided ideal 𝔞_{λ} := 𝕂[S_{r}] · e_{λ} from the isotypic decomposition 𝕂[S_{r}] = ⨁_{μ⊢r} 𝔞_{μ} of the group ring 𝕂[S_{r}] into minimal two-sided ideals 𝔞_{μ}. 𝔞_{λ} is that minimal two-sided ideal which contains all left ideals 𝕂[S_{r}] · y_{t} generated by Young symmetrizers y_{t} with Young frame λ. The idempotents e_{λ} fulfil

(3.29)  ∑_{λ⊢r} e_{λ} = id  ; e_{λ} · e_{λ′} = 0 if λ ≠ λ′ .

Theorem 3.15. Assume, that the symmetry class of a tensor field T ∈ T_{r}M, r ≥ 1, is defined by a primitive idempotent e ∈ 𝕂[S_{r}] whose representation σ according to Setting 3.6 satisfies σ ∼ [λ], λ ⊢ r. Then

(3.30)  ẽ = ∑_{{μ⊢r+1 \atop λ⊂μ}} h_{μ} , h_{μ} := ẽ · e_{μ}

yields the decomposition of ẽ into primitive idempotents h_{μ}, which corresponds to relation (3.17).

Proof. Because (3.17) is multiplicity-free, a decomposition of ẽ according to (3.17) into primitive idempotents contains exactly one primitive idempotent h_{μ} for every [μ] in (3.17). Every such h_{μ} lies in the corresponding two-sided ideal 𝔞_{μ}, i.e. h_{μ} ∈ 𝔞_{μ}. On the other hand, we can write ẽ = ẽ · id = ∑_{μ⊢r+1} ẽ · e_{μ}. Since ẽ · e_{μ} ∈ 𝔞_{μ}, we obtain (3.30). □

If we carry out calculations in large S_{r}, then a use of formula (3.30) leads to very high costs in calculation time and computer memory. However, fast discrete Fourier transforms can help to solve this problem.

14See H. Boerner [1, Sec.III.3, III.4] and R. Merris [24, Sec.4], in particular Exercise 41 in [24, p.117]. See also B. Fiedler [8, Prop.II.1.47, Prop.I.1.8].

15Here ω̆ denotes the regular representation of S_{r}.

Definition 3.16. A discrete Fourier transform for S_{r} is an isomorphism

(3.31)  D : 𝕂[S_{r}] → ⨂_{λ⊢r} 𝕂^{d_{λ}×d_{λ}}

according to Wedderburn’s theorem which maps the group ring 𝕂[S_{r}] onto an outer direct product ⨂_{λ⊢r} 𝕂^{d_{λ}×d_{λ}} of full matrix rings 𝕂^{d_{λ}×d_{λ}}. We denote by D_{λ} the natural projections D_{λ} : 𝕂[S_{r}] → 𝕂^{d_{λ}×d_{λ}}.

Since the subrings (0, …, 0, 𝕂^{d_{λ}×d_{λ}}, 0, …, 0) correspond to the two-sided ideals 𝔞_{λ} under a discrete Fourier transform we obtain

Corollary 3.17. If a discrete Fourier transform D is known for 𝕂[S_{r+1}], then the idempotents h_{μ} in (3.30) can be calculated by

(3.32)  h_{μ} = D^{−1} ((0, …, D_{μ}(ẽ), …, 0)) .

A use of (3.32) in computer calculations is much more efficient than an application of (3.30). A very efficient algorithm of a fast Fourier transform for S_{r} was developped by M. Clausen and U. Baum (see [2, 3]). It is based on Young’s seminormal representation of S_{r}. Our Mathematica package PERMS [9] uses Young’s natural representation of S_{r} as discrete Fourier transform.

4. Do Young symmetrizers describe the symmetry classes of ∇S or ∇A?

Now we turn to the question wheter the symmetry classes of ∇S and ∇A can be characterized by Young symmetrizers. According to Proposition 3.5 and Theorem 3.12 the symmetry classes of ∇S and ∇A are defined by the idempotents

e_{s} = f_{s} + h_{s} , e_{a} = f_{a} + h_{a} .

The idempotents f_{s} and f_{a} are proportional to the Young symmetrizers of the Young frames (r + 1) ⊢ r + 1 or (1^{r+1}) ⊢ r + 1, respectively. Now we investigate the

Problem 4.1. Can we find Young tableaux t_{s}, t_{a} with frame (r, 1) ⊢ r + 1 or (2, 1^{r−1}) ⊢ r + 1, respectively, such that the idempotents e_{t_{s}} = μ_{t_{s}}y_{t_{s}}, e_{t_{a}} = μ_{t_{a}}y_{t_{a}} are generating idempotents of the minimal left ideals 𝔩′_{s} := 𝕂[S_{r+1}] · h_{s}, 𝔩′_{a} := 𝕂[S_{r+1}] · h_{a}? Here y_{t_{s}} and y_{t_{a}} are the Young symmetrizers^{16} of the Young tableaux t_{s}, t_{a} and μ_{t_{s}}, μ_{t_{a}} ≠ 0 are constants.

If the answer is ”yes”, then the symmetry classes of ∇S or ∇A are determined by new idempotents

(4.1)  ẽ_{s} = f_{s} + e_{t_{s}} , ẽ_{a} = f_{a} + e_{t_{a}} ,

which are completely built from Young symmetrizers.

16We define a Young symmetrizer y_{t} of a Young tableau t by the formula y_{t} := ∑_{p∈H_{t}} ∑_{q∈V_{t}} sign(q) p · q, where H_{t}, V_{t} are the groups of horizontal or vertical permutations of t, respectively.

We investigated this problem by computer calculations by means of the Mathematica package PERMS [9] in the groups S_{3}, S_{4}, S_{5}, i.e.in the cases r = 2, 3, 4. We obtained the following results:

• No Young symmetrizer idempotent e_{t} of a Young frame (r, 1) ⊢ r + 1 reproduces or annihilates h_{s} ∈ 𝕂[S_{r+1}], i.e. no relation h_{s} · e_{t} = h_{s} or h_{s} · e_{t} = 0 is satisfied.

• For h_{a} ∈ 𝕂[S_{r+1}] there are many Young symmetrizer idempotents e_{t} with Young frame (2, 1^{r−1}) ⊢ r + 1 which reproduce or annihilate h_{a}, i.e. which fulfil h_{a} · e_{t} = h_{a} or h_{a} · e_{t} = 0. In particular, the idempotent e_{t} of the lexicographically greatest standard tableau

(4.2)  t = 1 r + 1 ; 2 ; ⋮ ; r

reproduces h_{a} whereas the idempotents e_{t} of all other standard tableaux of (2, 1^{r−1}) annihilate h_{a}.

For the S_{3} and r = 2, we also verified these results by calculations by means of the packages PERMS [9] and Ricci [21] in which we checked the action of idempotents e_{t}^{∗} onto tensors with a symmetry given by h_{s} or h_{a}. Mathematica notebooks of all above calculations can be downloaded from my internet page [4]

  http://home.t-online.de/home/Bernd.Fiedler.RoschStr.Leipzig/pnbks.htm

We present tables of all Young tableaux, whose idempotents e_{t} reproduce or annihilate h_{a}, in the Appendix.

Now we present theorems which tell us that essential parts of the above computer results are valid for all r ≥ 2.

It is well-known that two idempotents e, f ∈ 𝕂[S_{r}] generate the same left ideal iff e · f = e and f · e = f. In the case of primitive idempotents e, f we have

Lemma 4.2. If e, f ∈ 𝕂[S_{r}] are primitive idempotents, then the equations e·f = e and f · e = f are equivalent.

Proof. Assume that e · f = e is valid. Then e is an element of the left ideal 𝔩 = 𝕂[S_{r}] · f and generates a non-vanishing subideal 𝔩′ = 𝕂[S_{r}] · e of 𝔩. Since e and f are primitive, the left ideals 𝔩 and 𝔩′ are minimal. Thus we obtain 𝔩 = 𝔩′. But then, it follows f · e = f because f belongs to the left ideal 𝔩′ generated by e. □

Theorem 4.3. Consider the idempotent h_{a} for an arbitrary r ≥ 2. Then the Young symmetrizer idempotent e_{t} of the lexicographically greatest standard tableau (4.2) with Young frame (2, 1^{r−1}) ⊢ r + 1 reproduces h_{a}, i.e. h_{a} · e_{t} = h_{a}. Furthermore the idempotents e_{t} of all other standard tableaux t with Young frame (2, 1^{r−1}) ⊢ r + 1 annihilate h_{a}, i.e. h_{a} · e_{t} = 0.

Proof. The groups H_{t} and V_{t} of horizontal/vertical permutations of the Young tableau t according to (4.2) fulfil H_{t} = ⟨(1 , r +1)⟩ and V_{t} = S̃_{r}. Thus we can write

(4.3)  e_{t} = μ_{t} ∑_{p∈H_{t}} ∑_{q∈V_{t}} sign(q) p · q = μ_{t} · r! {id + (1 , r + 1)} · e_{a} .

It holds^{17} e_{t} · f_{a} = 0 since e_{t} and f_{a} are proportional to Young symmetrizers of the different Young frames (2, 1^{r−1}) and (1^{r+1}). From this and (4.3) we obtain

e_{t} · h_{a} = e_{t} · (e_{a} − f_{a}) = e_{t} · e_{a} = e_{t} .

Because e_{t} and h_{a} are primitive idempotents, Lemma 4.2 yields h_{a} · e_{t} = h_{a}.

It is well-known^{18}: If y_{t_{1}} and y_{t_{2}} are Young symmetrizers of two standard tableaux t_{1}, t_{2} which possess the same Young frame, and t_{1} is lexicographically smaller^{19} than t_{2}, then they satisfy y_{t_{2}} · y_{t_{1}} = 0. Because (4.3) is built from the lexicographically greatest standard tableau t of (2, 1^{r−1}), we obtain

h_{a} · e_{t′} = h_{a} · e_{t} · e_{t′} = 0

for the idempotent e_{t′} of every other standard tableau of (2, 1^{r−1}) □

Theorem 4.4. Consider the idempotent h_{s} for an arbitrary r ≥ 2. Then it holds h_{s} · e_{t} ≠ h_{s} for all Young tableaux t with Young frame (r, 1) ⊢ r + 1.

Proof. A Young tableau t with a Young frame (r, 1) ⊢ r + 1 has a form

(4.4)  t = \begin{matrix}k & ⋆ & ⋆ & … & ⋆ \\ l \end{matrix} .

If we assume that the first column of such a tableau contains the numbers k and l as in (4.4), then the groups of horizontal or vertical permutations of t read

(4.5)  H_{t} = {p ∈ S_{r+1} | p(l) = l} =: (S_{r+1})_{l}
(4.6)  V_{t} = ⟨(k , l)⟩ .

First we consider the case, that the first column of (4.4) does not contain r + 1. In this case we have

y_{t} = ( ∑_{p∈(S_{r+1})_{l}} p) · {id − (k , l)} .

But because V_{t} ⊆ S̃_{r}, we obtain {id − (k , l)} · e_{s} = 0 and y_{t} · e_{s} = 0. Thus e_{t} = μ_{t}y_{t} does not lie in the left ideal 𝔩_{s} := 𝕂[S_{r+1}] · e_{s} and can not play the role of a generating idempotent of 𝔩′_{s} := 𝕂[S_{r+1}] · h_{s}. Consequently h_{s} · e_{t} ≠ h_{s}.

17See e.g. H. Boerner [1, p.98] or W. Müller [25, p.73]. See also B. Fiedler [8, Sec.I.3.1].

18See A. Kerber [19, Vol.240/p.73] or H. Boerner [1, p.101]. See also B. Fiedler [8, Sec.I.3.1].

19A tableau t_{2} is regarded as greater than a tableau t_{1} (of the same Young frame), if the simultaneous run through the rows of both tableaux from left to right and from top to bottom reaches earlier in t_{2} a number which is greater than the number on the corresponding place in t_{1}.

Netxt we investigate the case l = r + 1. In this case we have H_{t} = S̃_{r}, V_{t} = ⟨(k , r + 1)⟩,

(4.7) e_{t} = μ_{t} (∑_{p∈S̃_{r}} p) · {id − (k , r + 1)} = μ_{t} · r! e_{s} · {id − (k , r + 1)} ,

which leads to

h_{s} · e_{t} = (e_{s} − f_{s}) · e_{t} = e_{s} · e_{t} = e_{t} .

We decompose h_{s} and e_{t} into parts that correspond to the right cosets of S_{r+1} relative to S̃_{r}. Obviously ℜ := {(i , r + 1) | i = 1, …, r + 1} is a complete set of representatives of those right cosets. If we arrange the summands of f_{s} and h_{s} according to the decomposition of S_{r+1} into cosets we obtain

f_{s} = \frac{1}{(r + 1)!} ∑_{s∈ℜ}(∑_{p∈S̃_{r}} p) · s

and

h_{s} = e_{s} − f_{s} = \frac{r}{(r + 1)!} (∑_{p∈S̃_{r}} p) − \frac{1}{(r + 1)!} ∑_{{s∈ℜ \atop s≠id}}(∑_{p∈S̃_{r}} p) · s .

Thus h_{s} has a non-vanishing part in every right coset of S_{r+1} relative to S̃_{r}. From (4.7) we see that e_{t} has non-vanishing parts only in the right cosets S̃_{r} · (k , r + 1) and S̃_{r}. This leads to h_{s} ≠ e_{t} and h_{s} · e_{t} = e_{t} ≠ h_{s}.

Finally, we consider the case k = r + 1. If t is the tableau (4.4) with k = r + 1, then t′ := (l, r +1)∘t is a Young tableau^{20} (4.4) with l = r +1. A relation t′ = p∘t, p ∈ S_{r+1}, between Young tableaux leads to

H_{t′} = p ∘ H_{t} ∘ p^{−1} , V_{t′} = p ∘ V_{t} ∘ p^{−1} , y_{t′} = p · y_{t} · p^{−1} .

For the above tableaux t′ := (l, r + 1) ∘ t we obtain y_{t′} = −(l, r + 1) · y_{t} since y_{t} · (l, r + 1)^{−1} = −y_{t}.

Now, if we assume h_{s} · e_{t} = h_{s}, then it follows from Lemma 4.2

e_{t} · h_{s} = e_{t} ⇒ y_{t′} · h_{s} = y_{t′} ⇒ e_{t′} · h_{s} = e_{t′} ⇒ h_{s} · e_{t′} = h_{s} .

However, the last equation is a contradiction to our proof in the case l = r + 1. □

20A Young tableau t of S_{r} can be regarded a one-to-one mapping of the boxes of the Young frame of t onto the set {1, …, r}. t maps every box onto that number which was placed into the box. Then the composition p ∘ t of a Young tableau t and a permutation p ∈ S_{r} is a Young tableau again.

5. Use of ∇S and ∇A in generator formulas of algebraic covariant derivative curvature tensors

Now we return to the question whether tensors (1.10) can be used as generators U for algebraic covariant derivative curvature tensors in formulas (1.7). In [13] we proved

Theorem 5.1. Let us denote by S, A ∈ T_{2}V symmetric or alternating tensors of order 2 and by U ∈ T_{3}V covariant tensors of order 3 whose symmetry class T_{𝔯} is defined by a fixed minimal right ideal 𝔯 from the equivalence class characterized by (2, 1) ⊢ 3. We consider the following types τ of tensors

(5.1)  τ :  \begin{matrix}y_{t′}^{∗}(S ⊗ U) & , & y_{t′}^{∗}(U ⊗ S) & , \\ y_{t′}^{∗}(A ⊗ U) & , & y_{t′}^{∗}(U ⊗ A) & , \end{matrix}

where y_{t′} ∈ 𝕂[S_{5}] is the Young symmetrizer of the standard tableau

t′ = 1 3 5 2 4 .

Then for each of the above types τ the following assertions are equivalent:

(1) The vector space of algebraic covariant derivative curvature tensors ℜ′ ∈ T_{5}V is the set of all finite sums of tensors of the type τ considered.

(2) The right ideal 𝔯 is different from the right ideal 𝔯_{0} := f · 𝕂[S_{3}] with generating idempotent

(5.2)  f := \{\frac{1}{2} (id − (1 3)) − \frac{1}{6} y\} , y := ∑_{p∈S_{3}} sign(p) p .

From Theorem 5.1 we obtain easily

Theorem 5.2. Let ∇ be a torsion-free covariant derivative on the mannifold M and p ∈ M. Then Statement (1) of Theorem 5.1 holds for the algebraic covariant derivative curvature tensors ℜ′ ∈ T_{5}M_{p} if we take the tensors U from one of the tensor sets

(5.3)  U = h_{s}^{∗}(∇S)|_{p} = ∇S|_{p} − sym(∇S)|_{p}

or

(5.4)  U = h_{a}^{∗}(∇A)|_{p} = ∇A|_{p} − alt(∇A)|_{p} = ∇A|_{p} − dA|_{p}

formed from the whole of symmetric or alternating tensor fields S, A ∈ T_{2}M.

Proof. Let us denote by T_{e} the symmetry class T_{𝔯} that is defined by a right ideal 𝔯 = e · 𝕂[S_{r}] with generating idempotent e ∈ 𝕂[S_{r}]. Then Proposition 3.7 yields

T_{e_{s}} = {(∇S)|_{p} | S ∈ T_{2}M symmetric}
T_{e_{a}} = {(∇A)|_{p} | A ∈ T_{2}M alternating}

from which we obtain^{21}

T_{h_{s}} = {h_{s}(∇S)|_{p} | S ∈ T_{2}M sym.} = {∇S|_{p} − sym(∇S)|_{p} | S ∈ T_{2}M sym.}
T_{h_{a}} = {h_{a}(∇A)|_{p} | A ∈ T_{2}M alt.} = {∇A|_{p} − alt(∇A)|_{p} | A ∈ T_{2}M alt.} .

Now we have only to check that the idempotents

h_{s} = e_{s} − f_{s} , h_{a} = e_{a} − f_{a}

do not generate the right ideal 𝔯_{0}. We do this by verifying

(5.5)  f · h_{s} ≠ h_{s} , f · h_{a} ≠ h_{a} .

The fastes way would be a computer calculation by means of PERMS [9]. A calculation by hand has the starting point (3.11), (3.25), (5.2) and z := \frac{1}{2}(id − (1 3)). From the rules

• ”symmetrization + alternation = 0”

• ”alternation + alternation = alternation” we obtain immediately

y · f_{s} = 0 , y · e_{s} = 0 , z · f_{s} = 0
y · f_{a} = y , y · e_{a} = y , z · f_{a} = f_{a}

Furthermore we have the products

z · e_{s} = \frac{1}{4} \{[1, 2, 3] + [2, 1, 3] − [2, 3, 1] − [3, 2, 1]\}
z · e_{a} = \frac{1}{4} \{[1, 2, 3] − [2, 1, 3] + [2, 3, 1] − [3, 2, 1]\} .

This leads to

f · h_{s} = z · e_{s} = \frac{1}{4} \{[1, 2, 3] + [2, 1, 3] − [2, 3, 1] − [3, 2, 1]\}

and

f · h_{a} = z · e_{a} − z · f_{a} − \frac{1}{6} y · e_{a} + \frac{1}{6} y · f_{a}
= z · e_{a} − f_{a} − \frac{1}{6} y + \frac{1}{6} y
= z · e_{a} − f_{a} .

But we see from these results that (5.5) is valid because • h_{s} is a linear combination of 6 permutations and z·e_{s} has only 4 summands,

• z · e_{a} ≠ e_{a}. □

21Note that the idempotents e_{s}, f_{s}, h_{s}, e_{a}, f_{a}, h_{a} fulfil e_{s}^{∗} = e_{s}, f_{s}^{∗} = f_{s}, h_{s}^{∗} = h_{s}, e_{a}^{∗} = e_{a}, f_{a}^{∗} = f_{a}, h_{a}^{∗} = h_{a}.

Appendix

Using the Mathematica package PERMS [9] for r = 2, 3, 4 we found Young symmetrizer idempotents e_{t} = μ_{t}y_{t} with Young frames (2, 1^{r−1}) ⊢ r+1 which reproduce (h_{a} ·e_{t} = h_{a}) or annihilate (h_{a} ·e_{t} = 0) the idempotent h_{a} ∈ 𝕂[S_{r+1}] of the symmetric group S_{r+1} considered. Here we present complete lists of the Young tableaux t for which the e_{t} possess such a property.

2 tableaux for r = 2 such that h_{a} · e_{t} = h_{a}.

{1, 3}, {2, 3}
{2}     {1}

2 tableaux for r = 2 such that h_{a} · e_{t} = 0.

{1, 2}, {2, 1}
{3}     {3}

6 tableaux for r = 3 such that h_{a} · e_{t} = h_{a}.

{1, 4}, {1, 4}, {2, 4}, {2, 4}, {3, 4}, {3, 4}
{2}     {3}     {1}     {3}     {1}     {2}
{3}     {2}     {3}     {1}     {2}     {1}

12 tableaux for r = 3 such that h_{a} · e_{t} = 0.

{1, 2}, {1, 2}, {1, 3}, {1, 3}, {2, 1}, {2, 1}, {2, 3}, {2, 3},
{3}     {4}     {2}     {4}     {3}     {4}     {1}     {4}
{4}     {3}     {4}     {2}     {4}     {3}     {4}     {1}
{3, 1}, {3, 1}, {3, 2}, {3, 2}
{2}     {4}     {1}     {4}
{4}     {2}     {4}     {1}

24 tableaux for r = 4 such that h_{a} · e_{t} = h_{a}.

{1, 5}, {1, 5}, {1, 5}, {1, 5}, {1, 5}, {1, 5}, {2, 5}, {2, 5},
{2}     {2}     {3}     {3}     {4}     {4}     {1}     {1}
{3}     {4}     {2}     {4}     {2}     {3}     {3}     {4}
{4}     {3}     {4}     {2}     {3}     {2}     {4}     {3}
{2, 5}, {2, 5}, {2, 5}, {2, 5}, {3, 5}, {3, 5}, {3, 5}, {3, 5},
{3}     {3}     {4}     {4}     {1}     {1}     {2}     {2}
{1}     {4}     {1}     {3}     {2}     {4}     {1}     {4}
{4}     {1}     {3}     {1}     {4}     {2}     {4}     {1}
{3, 5}, {3, 5}, {4, 5}, {4, 5}, {4, 5}, {4, 5}, {4, 5}, {4, 5}
{4}     {4}     {1}     {1}     {2}     {2}     {3}     {3}
{1}     {2}     {2}     {3}     {1}     {3}     {1}     {2}
{2}     {1}     {3}     {2}     {3}     {1}     {2}     {1}

72 tableaux for r = 4 such that h_{a} · e_{t} = 0.

{1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 3}, {1, 3},
{3}     {3}     {4}     {4}     {5}     {5}     {2}     {2}
{4}     {5}     {3}     {5}     {3}     {4}     {4}     {5}
{5}     {4}     {5}     {3}     {4}     {3}     {5}     {4}
{1, 3}, {1, 3}, {1, 3}, {1, 3}, {1, 4}, {1, 4}, {1, 4}, {1, 4},
{4}     {4}     {5}     {5}     {2}     {2}     {3}     {3}
{2}     {5}     {2}     {4}     {3}     {5}     {2}     {5}
{5}     {2}     {4}     {2}     {5}     {3}     {5}     {2}
{1, 4}, {1, 4}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1},
{5}     {5}     {3}     {3}     {4}     {4}     {5}     {5}
{2}     {3}     {4}     {5}     {3}     {5}     {3}     {4}
{3}     {2}     {5}     {4}     {5}     {3}     {4}     {3}
{2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 4}, {2, 4},
{1}     {1}     {4}     {4}     {5}     {5}     {1}     {1}
{4}     {5}     {1}     {5}     {1}     {4}     {3}     {5}
{5}     {4}     {5}     {1}     {4}     {1}     {5}     {3}
{2, 4}, {2, 4}, {2, 4}, {2, 4}, {3, 1}, {3, 1}, {3, 1}, {3, 1},
{3}     {3}     {5}     {5}     {2}     {2}     {4}     {4}
{1}     {5}     {1}     {3}     {4}     {5}     {2}     {5}
{5}     {1}     {3}     {1}     {5}     {4}     {5}     {2}
{3, 1}, {3, 1}, {3, 2}, {3, 2}, {3, 2}, {3, 2}, {3, 2}, {3, 2},
{5}     {5}     {1}     {1}     {4}     {4}     {5}     {5}
{2}     {4}     {4}     {5}     {1}     {5}     {1}     {4}
{4}     {2}     {5}     {4}     {5}     {1}     {4}     {1}
{3, 4}, {3, 4}, {3, 4}, {3, 4}, {3, 4}, {3, 4}, {4, 1}, {4, 1},
{1}     {1}     {2}     {2}     {5}     {5}     {2}     {2}
{2}     {5}     {1}     {5}     {1}     {2}     {3}     {5}
{5}     {2}     {5}     {1}     {2}     {1}     {5}     {3}
{4, 1}, {4, 1}, {4, 1}, {4, 1}, {4, 2}, {4, 2}, {4, 2}, {4, 2},
{3}     {3}     {5}     {5}     {1}     {1}     {3}     {3}
{2}     {5}     {2}     {3}     {3}     {5}     {1}     {5}
{5}     {2}     {3}     {2}     {5}     {3}     {5}     {1}
{4, 2}, {4, 2}, {4, 3}, {4, 3}, {4, 3}, {4, 3}, {4, 3}, {4, 3}
{5}     {5}     {1}     {1}     {2}     {2}     {5}     {5}
{1}     {3}     {2}     {5}     {1}     {5}     {1}     {2}
{3}     {1}     {5}     {2}     {5}     {1}     {2}     {1}

Acknowledgement. I would like to thank Prof. P. B. Gilkey for suggesting these investigations and for many important and helpful discussions.

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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