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I want to define a tensor that has the first two symmetries of Riemann tensor or maybe the last two. The symmetries of Riemann tensor are: 1) $ R_{\alpha\beta\gamma\lambda}=R_{\gamma\lambda\alpha\beta}$ 2)$R_{\alpha\beta\gamma\lambda}=-R_{\alpha\beta\lambda\gamma}$ and $R_{\alpha\beta\gamma\lambda}=-R_{\alpha\beta\lambda\gamma}$ 3)$R_{\alpha \beta \gamma \lambda} +R_{\alpha\lambda\beta\gamma} + R_{\alpha\gamma\lambda \beta }=0$ Could you help me to learn to define a tensor with mixed symmetries?

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1 Answer 1

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The skew symmetries and the interchange symmetry can be implemented easily by specifying them as symmetry generators in SymmetrizedArray:

rt=SymmetrizedArray[pos_:>Subscript["R",Row[pos]],{4,4,4,4},{
   {{2,1,3,4},-1},(* Skew symmetry in first two slots *)
   {{1,2,4,3},-1},(* Skew symmetry in last two slots *)
   {{3,4,1,2},1}(* Interchange symmetry between first and last pair of slots*)
}]
SymmetrizedArrayRules[%]
Length[%]

returns

result

This seems to work fine: one can check the symmetries by looking at the components and the number of components is also correct: if I recall my General Relativity lectures correctly the Riemann tensor has 21 independent components in four dimensions (20 when including the first (algebraic) Bianchi identity $R_{\alpha \beta \gamma \lambda} +R_{\alpha\lambda\beta\gamma} + R_{\alpha\gamma\lambda \beta }=0$).

The first (algebraic) Bianchi identity can not be implemented easily in terms of symmetry generators but one could manually select a number of components ($\frac{1}{24} d (d-1) (d-2) (d-3)$ where $d$ is the dimension of the tensor (slots)) to be expressed in terms of two other components. Depending on your dimension $d$ this might make sense or not. In $d<3$ the identity gives no additional symmetry, in $d=4$ just one, and so it only looks profitable for larger $d$.

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