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I would like to use TensorReduce to work on the tensor contraction $y_{ib}y_{ia}$ (where index $i$ is summed). However, TensorReduce does not bring the following t1 and t2 into a unique form:

$Assumptions = y ∈ Arrays[{d, d}];
t1 = TensorContract[TensorProduct[y, y], {{1, 3}}];
t2 = TensorTranspose[TensorContract[TensorProduct[y, y], {{1, 3}}], {2, 1}];
TensorReduce[t1]
TensorReduce[t2]

(* Output: *)
(* t1 -> TensorContract[y ⊗ y, {{1, 3}}] *)
(* t2 -> TensorTranspose[TensorContract[y ⊗ y, {{1, 3}}], {2, 1}] *)

Shouldn't they be brought into the same form? Thank you!

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up vote 2 down vote accepted

Symbolic tensor analysis is new in version 9 and it is not yet fully implemented.

For example, Mathematica knows that t1 and t2 are symmetric

TensorSymmetry[t1]
TensorSymmetry[t2]
(* Symmetric[{1, 2}] *)
(* Symmetric[{1, 2}] *)

But it can't simplify transposition automatically. I think you can contact to the Wolfram support. Now you can use the following workaround:

transpose[t_, n_: {2, 1}] := Module[{t1}, 
  TensorReduce[Transpose[t1, n], 
    Assumptions -> {t1 ∈ Arrays[TensorDimensions[t], Complexes, TensorSymmetry[t]]}] /. 
   t1 -> t]

transpose[t1]
(* TensorContract[y\[TensorProduct]y, {{1, 3}}] *)

It is interesting that everything works fine for the contraction of the right indexes $y_{ai}y_{bi}$:

$Assumptions = y ∈ Arrays[{d, d}];
t1 = TensorContract[TensorProduct[y, y], {{2, 4}}];
t2 = TensorTranspose[
   TensorContract[TensorProduct[y, y], {{2, 4}}], {2, 1}];
TensorReduce[t1]
TensorReduce[t2]
(* TensorContract[y\[TensorProduct]y, {{2, 4}}] *)
(* TensorContract[y\[TensorProduct]y, {{2, 4}}] *)
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