# Contracted indices can be interchanged?

I am trying to do some research in Lorentz transformations and I get a tensor of the form

$$M_{abcd}=\epsilon_{ab\mu \nu}\Lambda^\mu\hspace{0.1cm}_c\Lambda^\nu\hspace{0.1cm}_d$$ Where $$\epsilon_{ab\mu \nu}$$ is the totally antisymmetric (pseudo-)tensor of Levi-Civita.
Is it posible to interchange contracted indices so that I can get $$M_{abdc}=\epsilon_{ab\mu \nu}\Lambda^\mu\hspace{0.1cm}_d \Lambda^\nu\hspace{0.1cm}_c=-\epsilon_{ab \nu\mu}\Lambda^\mu\hspace{0.1cm}_d\Lambda^\nu\hspace{0.1cm}_c=-M_{abcd}$$.
Also is posible to check if $$M_{abcd}=uM_{cdab}$$ with $$u=\pm 1$$

• Is this a question about the software Mathematica or about the mathematics? – flinty Jul 13 at 13:10
• Note that if this were true then $2M_{abcd}=0$ so $M_{abcd}=0$. – Jackozee Hakkiuz Jul 15 at 1:06

A Wolfram Language approach to verifying your identity is to give assumptions on your tensors:

\$Assumptions = ϵ ∈ Arrays[{4,4,4,4}, Reals, Antisymmetric[{1,2,3,4}]] && Λ ∈ Matrices[{4,4}];


And then to use TensorReduce:

TensorReduce[TensorContract[TensorProduct[ϵ,Λ,Λ],{{3,5},{4,7}}]]
TensorReduce[-TensorContract[TensorProduct[ϵ,Λ,Λ],{{3,7},{4,5}}]]


TensorContract[TensorProduct[ϵ,Λ,Λ],{{3,5},{4,7}}]

TensorContract[TensorProduct[ϵ,Λ,Λ],{{3,5},{4,7}}]

Both expressions reduce to the same tensor.