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Why doesn't Simplify and FullSimplify operate on object generated using SymmetrizedArray?

For example:

FullSimplify[SymmetrizedArray[metric.riemann, {n, n, n, n}, Antisymmetric[{3, 4}]

leaves expressions like 1/2*(-2a'[t]^2+2a[t]*a''[t]) without obvious simplification of factors 2.

More transparent example: (FullSimplify acting on structured array and on normal array)

In:

AA = SymmetrizedArray[{{1, 2} -> (a^2 - b^2)^3, {1, 3} -> ((a-b) (a + b))^3, {2, 3} -> x}, {3, 3}, Antisymmetric[{1, 2}]];
B1 = Normal[FullSimplify[AA]]
B2 = FullSimplify[Normal[AA]]

Out:

{{0,(a^2-b^2)^3,(a-b)^3 (a+b)^3},{-(a^2-b^2)^3,0,x},{-(a-b)^3 (a+b)^3,-x,0}}
{{0,(a^2-b^2)^3,(a^2-b^2)^3},{-(a^2-b^2)^3,0,x},{-(a^2-b^2)^3,-x,0}}

Of course my expressions are much more complicated and time consuming. I can't do it by transforming a structured array to normal form, because it is better to do operations (in this case simplification) only on independent components (in order to economize computing time and memory).

Perhaps I should use Map, but I don't understand how it works with a structured array.

Any suggestions ?

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You main problem here is that a SymmetrizedArray is not an expression once you have constructed it. It is an atom, a raw object that (although it looks otherwise) cannot be decomposed into smaller expressions

AA = SymmetrizedArray[{{1, 2} -> (a^2 - b^2)^3, 
  {1, 3} -> ((a - b) (a + b))^3, {2, 3} -> x}, {3, 3}, 
  Antisymmetric[{1, 2}]];
AtomQ[AA]

(* True *)

This means, although it looks like it is a composed expression

InputForm[AA]
(* StructuredArray[SymmetrizedArray, {3, 3}, StructuredArray`StructuredData[
  SymmetrizedArray, {{1, 2} -> (a^2 - b^2)^3, 
   {1, 3} -> (a - b)^3*(a + b)^3, {2, 3} -> x}, Antisymmetric[{1, 2}]]] *)

You cannot take parts of it and e.g. AA[[1]] doesn't work. Neither does replacing or (as you planned) mapping a function over it work.

The only possible solution I see is to simplify your expressions before you build your structured array.

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    $\begingroup$ This is an explanation why it works in unexpected way. Thank you. Nevertheless, I realize that there is a certain solution and way to reach the goal. At first it is convenient to summarize the problem in simple form: "take a StructuredArray (with some symmetry) -> do a simplification only on independant components -> return simplified StructuredArray with respective symmetries". And that is a solution: BB = SymmetrizedArray[ FullSimplify[SymmetrizedArrayRules[AA]], TensorDimensions[AA], TensorSymmetry[AA]]. $\endgroup$ – Druid Apr 16 '15 at 9:58
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    $\begingroup$ Moreover this code have another advantage: it may be effectively parallelized: BB = SymmetrizedArray[ParallelMap[FullSimplify,SymmetrizedArrayRules[AA],{1}], TensorDimensions[AA], TensorSymmetry[AA]]. There is no problem with parallel processing of deeper levels of table/array, because already they are all in form of list of rules. $\endgroup$ – Druid Apr 16 '15 at 10:05
  • $\begingroup$ Right and it is exactly what I meant by * simplify your expressions before you build your structured array*. Btw, please verify carefully whether the ParallelMap is really faster than simply calling FullSimplify on the complete expression. $\endgroup$ – halirutan♦ Apr 16 '15 at 10:59
  • $\begingroup$ @Druid Thanks for the very handy function you came up with to simplify structured arrays. I use it so much that I set up two buttons for it (one to Simplify and one to FullSimplify) in a custom palette for working with tensors. I have referred to your function in the asked and answered question 'Conveniently solving tensor equations in which the contained tensors have various symmetries' (link). $\endgroup$ – Bill N Dec 6 '16 at 13:23

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