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For example dddg, the third derivative of the metric tensor, in dimension $4$.

I need to state that it is symmetric in the first two indices and symmetric in the last three indices.

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  • $\begingroup$ What do you mean by "state" here, and how will you be using the statement? I think how exactly you'll want to state it depends a lot on how you plan to use the statement! $\endgroup$
    – thorimur
    May 5 '20 at 7:52
  • $\begingroup$ I want to use it as an Assumption and TensorReduce an abstract tensor equation. $\endgroup$ May 5 '20 at 7:54
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From a small note in the documentation for Arrays,

The symmetry sym can be given in several forms. First, it can be given as expressions like Symmetric[{s_i, ..., s_k}] or Antisymmetric[{s_i,...,s_k}], with the slots s_i being different positive integers between 1 and the rank r. It can also be given as a list of generators of the form {perm,\[Phi]}, representing that the array stays invariant under simultaneous transposition by the permutation perm and multiplication by the root of unity \[Phi]. In addition, it can be given as the internal direct product {sym_1, sym_2, ...} of those forms.

So, we should simply be able to do

$Assumptions = {dddg ∈ Arrays[{4,4,4,4,4}, {Symmetric[{1, 2}], Symmetric[{3,4,5}]}

And indeed, TensorTranspose[dddg, {2, 1, 4, 5, 3}] // TensorReduce, for example, yields dddg.

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