On the number of terms in the Lovelock products
Abstract
In this short note we wonder about the explicit expression of the expanding of the pth Lovelock product. We use the 1990s’ works of S. A. Fulling et al. on the symmetries of the Riemann tensor, and we show that the number of independent scalars appearing in this expanding is equal to the number of Young diagrams with all row lengths even in the decomposition of the pth plethysm of the Young diagram representing the symmetries of the Riemann tensor.
1 Introduction
Lovelock theories are a set of modified gravity theories that can be seen as generalisations of General Relativity (GR) in higher dimension. They could have interesting cosmological implications (see [1, 2, 3, 4, 5]) or connections with string/Mtheories in which higherorder curvature terms appear naturally (see [6]).
They can be represented in the form of an action by a sum of scalar contractions of multiple copies of the Riemann curvature tensor. The contraction of p copies is called the pth Lovelock product. In even dimension, the nonvanishing term of highestdegree coincides to the Gauss–Bonnet–Chern scalar of the spacetime manifold, hence exhibits a promising relation with geometry.
Usually the Lovelock products are written as a product, and are handled in this form. This product is expanded only for small degrees, like \(p = 1\), 2 or 3. The astonishing complexity of the expanding for \(p=3\) in comparison to \(p = 1\) or 2 discourages to continue the expanding for further degrees. However the question is worth asking: what is the explicit formulation for the expanding of the general pth Lovelock product? The only terms appearing in such a development are the Riemann tensor, the Ricci tensor and the scalar curvature of the spacetime. But how many are they? And how are they combined? This is the topic of the present note. We do not solve entirely the problem, but give an answer to one of the questions: the number of independent scalars appearing in the expanding of the pth Lovelock product.
The answer we bring was in fact contained in a 25 years old paper from Fulling et al. [7]. This field of research finds its origin in computational aspects of the heat kernel in the context of quantum field theory and gravitation. Algorithms to simplify tensor calculus were developed, such as for the Computer Algebra System (CAS) REDUCE (see [8]) or Mathematica (see [9]).
Apparently we are the first one to make an explicit connection with Lovelock theories.
2 Notations
We represent the spacetime by a Lorentzian manifold \(({\mathcal {V}},g_{\mu \nu })\) of dimension \(n+1\), \(n \in {\mathbb {N}}\) standing for the spatial dimension. We choose \(c=\kappa =1\) for unit and \((1,+n)\) for the signature of \(g_{\mu \nu }\). We note
D  the LeviCivita connection of \(({\mathcal {V}},g_{\mu \nu })\), 
\(\mathrm {R}_{\mu \nu \rho \sigma }\)  the Riemann tensor of D, 
\(\mathrm {R}_{\mu \rho }\)  the Ricci tensor of D, 
\(\mathrm {R}\)  the curvature scalar of D, 
\({\text {d}}v = \sqrt{g}{\text {d}}^{n+1} x\)  the volume element of \({\mathcal {V}}\). 
\({\mathbb {R}}_p = \dfrac{1}{2^p}\delta _{\alpha _1 \beta _1 \alpha _2 \beta _2 \cdots \alpha _p \beta _p}^{\gamma _1 \delta _1 \gamma _2 \delta _2 \cdots \gamma _p \delta _p}\mathrm {R}_{\gamma _1 \delta _1}^{\alpha _1 \beta _1}\mathrm {R}_{\gamma _2 \delta _2}^{\alpha _2 \beta _2} \ldots \mathrm {R}_{\gamma _p \delta _p}^{\alpha _p \beta _p}\) the pth Lovelock product. 
\({\mathbb {R}}_0 = 1\), 
\({\mathbb {R}}_1 = \mathrm {R}\) is the scalar curvature, 
\({\mathbb {R}}_2 = \mathrm {R}^2  4 \mathrm {R}_\alpha ^\gamma \mathrm {R}^\alpha _\gamma + \mathrm {R}_{\alpha \beta }^{\gamma \delta }\mathrm {R}^{\alpha \beta }_{\gamma \delta }\) corresponds to the Gauss–Bonnet term for \(n+1=4\), 
\(\begin{aligned} {\mathbb {R}}_3 =&\, \mathrm {R}^3 +2\mathrm {R}_{\alpha \beta }^{\gamma \delta }\mathrm {R}_{\gamma \delta }^{\varepsilon \eta }\mathrm {R}_{\varepsilon \eta }^{\alpha \beta } +3\mathrm {R}\mathrm {R}_{\alpha \beta }^{\gamma \delta }\mathrm {R}_{\gamma \delta }^{\alpha \beta } \\&+8\mathrm {R}_{\alpha \beta }^{\gamma \eta }\mathrm {R}_{\gamma \delta }^{\varepsilon \beta }\mathrm {R}_{\varepsilon \eta }^{\alpha \delta } 12\mathrm {R}\mathrm {R}_\alpha ^\beta \mathrm {R}_\beta ^\alpha +16\mathrm {R}_\alpha ^\beta \mathrm {R}_\beta ^\gamma \mathrm {R}_\gamma ^\alpha \\&24\mathrm {R}_\alpha ^\beta \mathrm {R}_{\gamma \delta }^{\alpha \varepsilon }\mathrm {R}_{\beta \varepsilon }^{\gamma \delta } +24\mathrm {R}_\alpha ^\beta \mathrm {R}_\gamma ^\delta \mathrm {R}_{\beta \delta }^{\alpha \gamma }\end{aligned}\), and so on, until 
\({\mathbb {R}}_p = 0\) for \(p > p_n\), because of the antisymmetries of \(\mathrm {R}_{\alpha \beta }^{\gamma \delta }\). 
3 Young diagrams
Even though a computer can deal with this computation, the quick explosion of the number of terms puts the question of an explicit formula for the expanding of the \({\mathbb {R}}_p\)’s.
For all \(k \ge 1\), every ktensor \(T_{a_1 \ldots a_k}\) can be mapped to a representation of \({\mathcal {S}}_k\). Just as this representation can be decomposed into irreducible representations of \({\mathcal {S}}_k\) encoded by Young diagrams of size k, the tensor \(T_{a_1 \ldots a_k}\) can be decomposed as well on a basis \({\mathcal {T}}_k\) of ktensors corresponding to Young diagrams of size k. More precisely, the tensors of \({\mathcal {T}}_k\) have peculiar symmetries which are encoded in standard Young tableaux of size k. We note with \(\longleftrightarrow \) this correspondence.
Intuitively, contracting a pair of indices of the tensors can be understood as crossing a pair of cells off the Young diagram. If the two cells lie in different rows, the result vanishes because of the antisymmetrisation between the rows. If the two cells lie in the same row, the contraction is nontrivial. Hence, all rows must have even lengths so that the resulting empty diagram correspond to a nontrivial scalar.
In case \(U_{a_1 \ldots a_k} = U'_{b_1 \ldots b_l}\), the list of irreducible representations is restricted by the symmetries under the exchange of the two tensors: this is not an outer product anymore, but a new operation called a plethysm, \(\otimes \). This is the case we are interested in.
4 Plethysms of the Riemann tensor

two rows, it represents a scalar involving only the scalar curvature \(\mathrm {R}\);

three rows, it represents a scalar involving \(\mathrm {R}\) and the Ricci tensor \(\mathrm {R}_{\mu \nu }\);

four rows or more, it represents a scalar involving \(\mathrm {R}\), \(\mathrm {R}_{\mu \nu }\) and the Riemann tensor \(\mathrm {R}_{\mu \nu }^{\rho \sigma }\).
For the rest, there is no formula. Yet there exist algorithms able to deal with the computation. One can find such algorithms and applications for Maple in [10, 11, 12, 13]. There are also algorithms for the language REDUCE, [14], or Java, [15]. Algorithms for tensor simplification in Mathematica (package Tools of Tensor Calculus) can be found in [16, 17]. The program Cadabra can be used as well.
5 Conclusion
Notes
Acknowledgements
I would like to thanks Professor Stephen Fulling for having made himself available and for the useful references he gave to me.
References
 1.S. Nojiri, S.D. Odintsov, M. Sasaki, Gauss–Bonnet dark energy. Phys. Rev. D 71, 123509 (2005)ADSCrossRefGoogle Scholar
 2.S. Nojiri, S.D. Odintsov, Modified Gauss–Bonnet theory as gravitational alternative for dark energy. Phys. Lett. B 631, 1–6 (2005)ADSMathSciNetCrossRefGoogle Scholar
 3.S. Nojiri, S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy. eConf C0602061, 06 (2006)Google Scholar
 4.S. Nojiri, S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Methods Mod. Phys. 4, 115 (2007)MathSciNetCrossRefGoogle Scholar
 5.G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, S. Zerbini, Dark energy in modified Gauss–Bonnet gravity: latetime acceleration and the hierarchy problem. Phys. Rev. D 73, 084007 (2006)ADSCrossRefGoogle Scholar
 6.T. Torii, H. Shinkai, \(N+1\) formalism in Einstein–Gauss–Bonnet gravity. Phys. Rev. D 78, 084037 (2008)ADSMathSciNetCrossRefGoogle Scholar
 7.S.A. Fulling, R.C. King, B.G. Wybourne, C.J. Cummins, Normal forms for tensor polynomials. I. The Riemann tensor. Class. Quantum Gravity 9(5), 1151 (1992)ADSMathSciNetCrossRefGoogle Scholar
 8.A.A. Bel’kov, A.V. Lanyov, A. Schaale, Calculation of heatkernel coefficients and usage of computer algebra. Comput. Phys. Commun. 95(2), 123–130 (1996)ADSCrossRefGoogle Scholar
 9.M.J. Booth, HeatK: a mathematica program for computing heat kernel coefficients. High Energy Phys. Theory (1998). arXiv:hepth/9803113
 10.R. Portugal, An algorithm to simplify tensor expressions. Comput. Phys. Commun. 115(2), 215–230 (1998). Computer Algebra in Physics ResearchADSMathSciNetCrossRefGoogle Scholar
 11.R. Portugal, Algorithmic simplification of tensor expressions. J. Phys. A Math. Gen. 32(44), 7779 (1999)ADSMathSciNetCrossRefGoogle Scholar
 12.R. Portugal, B.F. Svaiter, Grouptheoretic approach for symbolic tensor manipulation: I. Free indices. Math. Phys. (2001). arXiv:mathph/0107031
 13.L.R.U. Manssur, R. Portugal, B.F. Svaiter, Grouptheoretic approach for symbolic tensor manipulation: II. Dummy indices. Int. J. Mod. Phys. C 13(07), 859–879 (2002)ADSCrossRefGoogle Scholar
 14.V.A. Ilyin, A.P. Kryukov, ATENSOR—REDUCE program for tensor simplification. Comput. Phys. Commun. 96, 36–52 (1996)ADSCrossRefGoogle Scholar
 15.V. Ilyin, A. Kryukov, A. Rodionov, G. Shpiz, The use of group and algebraic properties of tensor expressions in an objectoriented CA system. Programm. Comput. Softw. 26(1), 39–41 (2000)CrossRefGoogle Scholar
 16.A. Balfagon, X. Jaen, Simplifying tensor polynomials with indices. Gen. Relativ. Quantum Cosmol. (1998). arXiv:grqc/9809022
 17.A. Balfagon, X. Jaen, Nondimensional simplification of tensor polynomials with indices. Gen. Relativ. Quantum Cosmol. arXiv:grqc/9912062 (1999)
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