Reordering indices in alphabetical order

[Sukruti]

Hello,

Does anyone know how to make Mathematica reorder the indices of a given tensorial expression with free indices, in ascending alphabetical order?

Attached notebook shows the question more explicitly.

Best,

Sukruti

[Thomas Bäckdahl]

Hi!

This was a bit more complicated than I thought because FindFreeIndices sorts the indices, so it can not be used to extract the index ordering. However, FindIndices keeps the order of the indices.

The following two functions might do what you need. SortFreeInds can be dangerous, in particular if you have indices belonging to different VBundles. In that case you have to separate the indices into groups belonging to the same VBundle and only sort within each group. I have also ignored the possibility of blocked indices.

FreeIndsDown[expr_] := 
 ReplaceIndex[
  Evaluate@expr, (# -> DownIndex[#]) & /@ 
   List @@ FindFreeIndices[Evaluate@expr]]

SortFreeInds[expr_] := 
 ReplaceIndex[Evaluate@expr, 
  Thread[(# -> Sort@#) &@
      DeleteCases[List @@ #, 
       Alternatives @@ xAct`xTensor`Private`TakeEPairs[#]] &@
    FindIndices[Evaluate@expr]]]

Regards
Thomas

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[Sukruti]

Dear Thomas,

Thank you very much for defining the two functions.

Unfortunately the first one, FreeIndsDown[], does not seem to work as needed. It sends all the free indices down but it is not reordering them. I wonder what the reason could be.

Would it be easier to define a sorting function for terms which already have all their free indices either up or down?

The second function SortFreeInds[] works as needed. I will keep in mind that it should be used only for indices belonging to one vector bundle.

Thanks again for your help.

Best,

Sukruti

[Thomas Bäckdahl]

Hi!

The first function is not supposed to sort the indices, just move them down.
The idea was that if you want to identify expressions regardless of ordering of indices and index positions, you use both functions, one after the other. If you want to keep the index positions, but sort the free indices, you just use the second function.

Regards
Thomas

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[Sukruti]

Hi,

Oh I see, FreeIndsDown[] is not supposed to sort the indices. In that case, both functions are indeed working as intended.

It's true my aim is to identify expressions regardless of the ordering and positions of indices, and for this purpose the two functions you defined are useful when applied one after the other.

Since FindFreeIndices[] sorts the free indices, I initially thought it might also be usable for sorting the indices of a given expression, similar to what SortFreeInds[] does. But I haven’t yet managed to define a sorting function based on FindFreeIndices[]. It’s not necessary right now since I’m working with only one VBundle, for which SortFreeInds[] suffices. Still, it’s an interesting observation that although FindFreeIndices[] returns indices in sorted order, it’s not obvious how to leverage that for defining the desired sorting function.

Best,
Sukruti