Methods for the construction of generators of algebraic curvature tensors

We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors. In [10, 12, 13] we constructed and investigated generators of algebraic curvature tensors and algebraic covariant derivative curvature tensors. These investigations followed the example of the paper [14] by S.A. Fulling, R.C. King, B.G.Wybourne and C.J. Cummins and applied tools from Algebraic Combinatorics such as Young tableaux, symmetry operators (in particular Young symmetrizers), the LittlewoodRichardson rule, but also discrete Fourier transforms of symmetric groups. The present paper is a short summary of [10, 12, 13] in which we want to demonstrate the use of these methods. 1. The problem Let V be a finite dimensional K-vector space, K = R,C, and let TrV denote the K-vector space of covariant tensors of order r over V . DEFINITION 1. Algebraic curvature tensors R ∈ T4V and algebraic covariant derivative curvature tensors R ∈ T5V are tensors of order 4 or 5 whose coordinates satisfy Rijkl = −Rjikl = Rklij R ′ ijklm = −R ′ jiklm = R ′ klijm Rijkl + Riklj + Riljk = 0 R ′ ijklm + R ′ ikljm + R ′ iljkm = 0 R′ijklm + R ′ ijlmk + R ′ ijmkl = 0 . They are tensors which possess the same symmetry properties as the Riemannian curvature tensor Rijkl and its covariant derivative Rijkl;m of a Levi-Civita connection ∇. ∗ 1991 Mathematics Subject Classification: 53B20, 15A72, 05E10, 16D60, 05-04. [Author and title] 2 The vector space of algebraic curvature tensors R ∈ T4V is spanned by each of the following sets of tensors (P. Gilkey [16, pp.41-44], B. Fiedler1 [9]) γ(S)ijkl := SilSjk − SikSjl , S symmetric (1) α(A)ijkl := 2AijAkl +AikAjl −AilAjk , A skew − symmetric . (2) The vector space of algebraic covariant derivative curvature tensors R ∈ T5V is spanned by the following set of tensors (P. Gilkey [16, p.236], B. Fiedler [10]) γ̂(S, Ŝ)ijkls := SilŜjks − SjlŜiks + SjkŜils − SikŜjls , S , Ŝ symmetric. (3) PROBLEM 2. In the present paper we search for generators of algebraic curvature tensors R or algebraic covariant derivative curvature tensors R which can be formed by a suitable symmetry operator from the following types of tensors R : U ⊗w , U ∈ T3V , w ∈ T1V , (4) R : U ⊗W , U ∈ T3V , W ∈ T2V , (5) where W and U belong to symmetry classes of T2V and T3V which are defined by minimal right ideals r ⊂ K[S2] and r̂ ⊂ K[S3], respectively. We use Boerner’s definition [1, p.127] of symmetry classes of tensors. An element a = ∑ p∈Sr a(p)p ∈ K[Sr] of the group ring of the symmetric group Sr can be considered a symmetry operator for covariant tensors T ∈ TrV . The action of a on T is defined by: (aT )(v1, . . . , vr) := ∑ p∈Sr a(p)T (vp(1), . . . , vp(r)) , vi ∈ V . (6) DEFINITION 3. Let r ⊆ K[Sr] be a right ideal of K[Sr] for which an a ∈ r and a T ∈ TrV exist such that aT 6= 0. Then the tensor set Tr := {aT | a ∈ r , T ∈ TrV } (7) is called the symmetry class of tensors defined by r. Boerner [1, p.127] showed: If e ∈ K[Sr] is a generating idempotent of r, i.e. r = e · K[Sr], then it holds T ∈ TrV belongs to Tr ⇔ eT = T . [9] uses also tools of Algebraic Combinatorics, in particular plethysms. [Author and title] 3 2. Young symmetrizers Young symmetrizers are important symmetry operators. In particular the symmetries of the Riemann tensor R and its covariant derivatives are characterized by a Young symmetrizer. First we define Young tableaux. A Young tableau t of r ∈ N is an arrangement of r boxes such that 1. the numbers λi of boxes in the rows i = 1, . . . , l form a decreasing sequence λ1 ≥ λ2 ≥ . . . ≥ λl > 0 with λ1 + . . .+ λl = r, 2. the boxes are fulfilled by the numbers 1, 2, . . . , r in any order. For instance, the following graphics shows a Young tableau of r = 16. λ1 = 5 11 2 5 4 12 λ2 = 4 9 6 16 15 λ3 = 4 8 14 1 7 λ4 = 2 13 3 λ5 = 1 10 