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Yi Wang
Posts:
33
Registered:
3/30/12


How does TensorReduce use assumptions?
Posted:
Jan 23, 2014 3:18 AM


I would like to use TensorReduce by assuming that certain patterns of functions are tensors. From documentation of TensorReduce:
"If TensorDimensions[ten] does not return a list of dimensions, then the expression ten is returned unchanged."
I would have inferred from above that if I modify TensorDimensions[ten], TensorReduce should work. Thus I did
Unprotect[TensorDimensions]; TensorDimensions[f_[g__]] := d & /@ {g}; Protect[TensorDimensions];
Assuming[ t \[Element] Arrays[{d, d}, Antisymmetric[All]] , TensorReduce @ TensorContract[ t\[TensorProduct]f[DN, DN], {1, 4}]]
However, this doesn't work. i.e. TensorReduce does nothing, and the result is
TensorContract[ t\[TensorProduct]f[DN, DN], {{1, 4}}]
To compare, having defined
Assuming[ t \[Element] Arrays[{d, d}, Antisymmetric[All]] && f[DN, DN] \[Element] Arrays[{d, d}], TensorReduce @ TensorContract[ t\[TensorProduct]f[DN, DN], {1, 4}]]
 TensorContract[ t\[TensorProduct]f[DN, DN], {{2, 4}}]
the result is indeed simplified as desired:
I also tried to modify TensorSymmetry, but without luck either.
I'd like to understand what are the assumptions that TensorReduce really uses. Is there a way that I can work with TensorReduce as above, with pattern like declaration of tensors?
PS: Currently I generate a list of assumptions of f_[g__] using Cases, and put those assumptions together in Assuming. This makes the code slow and ugly.



