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! $\endgroup$– thorimurCommented May 5, 2020 at 7:52
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$\begingroup$ I want to use it as an Assumption and TensorReduce an abstract tensor equation. $\endgroup$– user3257842Commented May 5, 2020 at 7:54
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1 Answer
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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}]orAntisymmetric[{s_i,...,s_k}], with the slotss_ibeing different positive integers between1and the rankr. 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.
