Raymond G. McLenaghan

Robert Milson

Roman G. Smirnov

Comptes Rendus Mathématique

1/phic to aense thatp. As anesduoupe destenons laemetrichasC. R. Acad. Sci. Paris, Ser. I 339 (2004) 621–624http://france.elsevier.com/direct/CRASSDifferential Geometry/Mathematical PhysicsKilling tensors as irreducible representationsof the general linear group✩Raymond G. McLenaghana, Robert Milsonb, Roman G. Smirnovba Department of Applied Mathematics, Universityof Waterloo, Waterloo, Ontario N2L 3G1, Canadab Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, CanadaReceived 6 July 2004; accepted 12 July 2004Available online 14 October 2004Presented by Yvonne Choquet-BruhatAbstractWe show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorcertain highest weight, irreducible representation of the general linear group. The isomorphism is equivariant in the sthe natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgrouapplication, we deduce the Delong–Takeuchi–T ompson formula on the dimension of the vector space ofKilling tensors fromthe classical Weyl dimension formula.To cite this article: R.G. McLenaghan et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméTenseurs de Killing comme des représentations irréductibles du groupe linéaire. Nous démontrons que l’espace dtenseurs de Killing d’un ordredonné est naturellement isomorphe à une représentation irréductible de plus haut poidsgroupe linéaire. L’isomorphisme est équivariant ; les transformations par isométries correspondent à l’inclusion du grisométries comme un sous-groupe particulier du groupe linéaire. Comme application de cet isomorphisme nous obformule de Delong–Takeuchi–Thompson sur la dimension de l’espace des tenseurs de Killing à partir de la formule classiqude dimension de Weyl.Pour citer cet article : R.G. McLenaghan et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. Introduction and preliminariesLet (Mn,gαβ) be ann-dimensional, pseudo-Riemannian manifold of constant curvature. Various geostructures onMn can be described in terms ofG, the corresponding Lie group of isometries [4,8]. Recently, it✩ R.G.M, R.M, and R.G.S. were supported in part by the National Sciences and Engineering Research Council of Canada.E-mail addresses:rgmclena@uwaterloo.ca (R.G. McLenaghan), milson@mathst .dal.ca (R. Milson), smirnov@mathstat.dal.ca(R.G. Smirnov).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.07.017622 R.G. McLenaghan et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 621–624sors [5].cesid for all.ling]. Thus,themor-efine aroup [5].tofws ofbyngbeen shown that a group and invariant-theoretic approach can also be applied to the study of Killing tenContinuing in the same direction, we announce a characteriza ion of Killing tensors on pseudo-Riemannian spaof constant curvature as certain irreducible representations of the general linear group. Our result is valdimensions, all signatures of the metric, and for all values (positive, negative, and zero) of the curvature.A Killing tensor of valencep is a symmetric tensor fieldhα1...αp satisfying∇(α0hα1...αp) = 0. (1)Let Kp denote the vector space of such tensor fields onMn, with K1 being the Lie algebra of Killing vector fieldsOn spaces of constant curvature every Killing tensor canbe represented as a sum of symmetric products of Kilvectors [2]. However, certain such products vanish identically; these form the so-called syzygy module [6Kp is a quotient of Symp K1 by the syzygy module. It is desirable to describe this quotient explicitly.Takeuchi took a step in this direction by showing [6] thatKp is isomorphic to a certain representation oflinear group by invoking the Bott–Borel–Weil theorem [1]. We go further by exhibiting an elementary isophism betweenKp and this irreducible representation. Our construction uses a Young symmetrizer to dcomplementary subspace to the syzygy module.The main result resolves an outstanding conjecture about the equivariance of the action of the isometry gAn additional application is a simple proof of the Delong–Takeuchi–Thompson dimension formula [2,6,7].2. Representation theory of the general linear groupIn this section we recall some basic facts from the representation theory of the general linear group [3]. LeVbe a finite-dimensional, real vector space,Sm the symmetric group onm elements, andQSm the group algebrawith rational coefficients. Letm = λ1 + · · · + λ, whereλ1  λ2  · · ·λ > 0, be a partition of an integerm.Let a = (a1, . . . , an), whereaj = λj − λj+1, be the corresponding weight sequence. In the sequelτij denotes thestandard tableau, defined byτij = λ1 + · · · + λi−1 + j , wherei = 1, . . . ,m is the row index andj = 1, . . . , λi isthe column index.Let SaV denote the irreducible representation of GL(V ) generated by the highest-weight elementSyma1 Λ1V ⊗ · · · ⊗ Syman ΛnV . Let ca ∈ QSm be the Young symmetrizer (i.e.,c2a = ca) defined byca = ka∑q,r(−1)qqr. (2)In the above summationq andr range over all permutations preserving, respectively, the columns, and the rothe tableau. The normalization constant is given byka = 1b1! · · ·b!∏1i<j(bi − bj ), with bj = λj +  − j. (3)The representationSaV consists of all tensors satisfyingcaA = A. The dimension of the representation is giventhe Weyl dimension formula:dimSaV =∏1i<jnλi − λj + j − ij − i . (4)We will be particularly interested in weights of the forma = (0,p,0, . . . ,0), and denote the correspondirepresentation, concisely, byS{p}V . The corresponding symmetry condition admits the following description. Fixpand letΞ = Λ if p is odd, andΞ = Sym if p is even. Letσ : Symp Λ2V → Ξ2 Symp V , be defined byσ(K)a1a2···apb1b2···bp =∑r∈SpKa1br(1)a2br(2)···apbr(d) , K ∈ Symp Λ2V, (5)and letκ :Ξ2 Symp V → Symp Λ2V , be defined byR.G. McLenaghan et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 621–624 623in-etrysists ofrdκ(S)a1b1a2b2···apbp = 2pŜ[a1b1][a2b2]···[apbp], (6)where we defineŜa1b1a2b2···apbp = Sa1a2···apb1b2···bp , S ∈ Ξ2 Symp V .Proposition 2.1. The representationS{p}V consists ofK ∈ Symp Λ2V satisfyingκ(σ(K)) = (p + 1)!K. (7)Thus, the tensors in questions havep skew-symmetric pairs of indices, but are symmetric with respect toterchange of the undisturbed pairs. Eq. (7) means that these tensors also satisfy a number of other symmconditions, e.g., the cyclic conditionKabc··· + Kbca··· + Kcab··· = 0. (8)Indeed, forp = 2, conditions (7) and (8) are equivalent. The corresponding irreducible representation contype (4,0) tensors having the symmetry type of the Riemann curvature tensor. However, forp > 2, Eq. (7) impliesadditional symmetry conditions, which we will not analyze here.Let us also note that a direct application of (4) withn = dimV − 1 givesdimS{p}V = 1n(n + pp + 1)(n + p − 1p). (9)3. Killing tensorsHenceforth, we assume dimV = n + 1. Let(u,v) = uava, u,v ∈ Vbe a signature(n + 1− q, q) inner product, which we use to endowV with a flat metric. We consider the standamodels [8] of constant-curvature submanifoldsι :Mn → V . These are either a generalized unit sphere,Mn = {x ∈ V : xaxa = 1}; (10)or a unitally displaced hyperplane,Mn = {x ∈ V : uaxa = 1}, uaua = 1. (11)Let G < GL(V ) be the corresponding group of orientation preserving isometries. In the first case,G ∼= SO(n+1−q, q). In the second case,G is isomorphic to the semi-direct product SO(n − q, q)  V .UseφA, A ∈ Symp V ∗ to denote the type(0,p) symmetric tensor field onV with constant components. LetπAbe the corresponding degreep, homogeneous polynomialπA(x) = A(x,x, . . . ,x), x ∈ V.UseφA⊗B, A,B,∈ Symp V ∗ to denote the type(0,p) symmetric tensor field defined byφA⊗B(x) = πA(x)φB, x ∈ V.SinceΞ2 Symp V ∗ is a subspace of Symp V ∗ ⊗ Symp V ∗, the above definition extends to give a type(0,p) tensorfield φS for everyS ∈ Ξ2 Symp V ∗. ForK ∈ Symp Λ2V ∗, useK̂ to denote the symmetric type(0,p) tensor fieldonMn defined byK̂ = ι∗(φσ(K)),624 R.G. McLenaghan et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 621–624ingorsed in[5], and–ndwhereι∗ denotes the inclusion pullback map, and whereσ is the symmetrization operator defined in (5). Treatthe vector space componentsxa, a = 1, . . . , n + 1 as functions of local coordinates onMn, and using∂αxa,α = 1, . . . , n to denote the corresponding partial derivatives, we haveK̂α1···αp(x) = Ka1b1···apbp (∂α1xa1) · · · (∂αpxap)xb1 · · ·xbp , x = (x1, . . . , xn+1) ∈ Mn ⊂ V.Proposition 3.1. Every sucĥK is a Killing tensor onMn.The above follows from (1), (10), (11) by local coordinate calculations.Proposition 3.2. The mappingΥ : Symp Λ2V ∗ →Kp defined byΥ (K) = K̂ is a linear surjection.The casep = 1 corresponds to the isomorphism betweenΛ2V ∗ andK1, and can be established directly. Fp > 1, we prove the proposition by noting that, on a spacesof constant curvature,all Killing tensors can begenerated using symmetric products of Killing vectors [2,6].Proposition 3.3. The kernel ofΥ is isomorphic to the kernel ofσ , as defined in Eq.(5).Let Adg, g ∈ GL(V ), denote the usual adjoint action onV ∗, namely(Adg u)(v) = u(g−1v), v ∈ V, u ∈ V ∗.We also let Adg denote the corresponding action ofg on the tensor algebra ofV ∗. For an isometryg ∈ G, thepull-back map(g−1)∗ is an invertible linear transformation ofKp , which defines a representation ofG onKp .Proposition 3.4. For g ∈ G, K ∈ Symp Λ2V ∗, we have(g−1)∗(K̂) = Âdg K.Here is our main theorem. It follows directly from Propositions 3.3 and 3.4. A form of this result, expresterms of the infinitesimal action on the parameter space induced by the isometry group, was conjectured inverified for a number of particular cases.Let Πp denote the restrictionΥ to S{p}V ∗.Theorem 3.5. The linear mapΠp :S{p}V ∗ →Kp is a G-representation isomorphism.As per the comment at the end of the preceding section,we also obtain a direct proof of the Delong–TakeuchiThompson formula [2,6,7].Corollary 3.6. The dimension ofKp is given by the right-hand side of Eq.(9).References[1] R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957) 203–247.[2] R. Delong, Killing tensors and the Hamilton–Jacobi equation, Ph.D. thesis, University of Minnesota, 1982.[3] W. Fulton, J. Harris, Represntation Theory, Springer, 1991.[4] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001.[5] R.G. McLenaghan, R.G. Smirnov, D. The, An extension of the classic l theory of algebraic invariants to pseudo-Riemannian geometry aHamiltonian mechanics, J. Math. Phys. 45 (2004) 1079–1120.[6] M. Takeuchi, Killing tensor fields on spaces of constant curvature, Tsukuba J. Math. 7 (1983) 233–255.[7] G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986) 2693–2699.[8] J. Wolf, Spaces of Constant Curvature, Publish or Perish, Houston, TX, 1984.

Killing tensors as irreducible representations of the general linear group

McLenaghan, Raymond G.

Milson, Robert

Smirnov, Roman G.

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2004.07.017.pdf

Killing tensors as irreducible representations of the general linear group

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