RE: [xAct: 1673] Independent tensors

[jo...@xact.es]

Hi,

I think I don't understand this question. Can you send examples of tensors that are different but still you consider them "repetitions"?

Do you use ToCanonical during the process of tabulating the possibilities, to remove equivalent cases?

Have you had a look at Thomas' SymManipulator package?

Cheers,

Jose.

-------- Original Message --------
Subject: [xAct: 1673] Independent tensors
From: Gaurav Narain <gaun...@gmail.com>
Date: Sun, January 17, 2016 8:51 am
To: xAct Tensor Computer Algebra <xa...@googlegroups.com>

I have the following product of metrics

g[-a,-b] g[-c,-d] g[-e,-f]

I want to construct all possible tensors using 3 metics which are symmetric is in (a,b), (c,d) and (e,f). for-exmaple the following 

combination (g[-a,-c]g[-b,-d]+g[-b,-c]g[-a,-d])g[-e,-f] has the required symmetris. 

Using indexconfiguration and symmetrisation I am able to tabulate them all, but there are many repetition.

I want to pick only the independent ones. Can anyone suggest a simple way of picking independent tensors only?

Gaurav.

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[Mario Herrero Valea]

HI,

Maybe the Young tableaux option in xTras is what you are looking for…

Cheers,

Mario 

[Thomas Bäckdahl]

Hi!

I am not sure of exactly what you want, but the following should produce a set of expressions of the kind you asked for. This takes into account the index symmetry groups, but not Young tableau symmetries. Much more powerful tools are implemented for spinors, but only in 4D spacetimes with signature + - - -.

<< xAct`SymManipulator`
$PrePrint = ScreenDollarIndices;
$DefInfoQ = False;
DefManifold[M4, 4, {a, b, c, d, e, f, h, l, m, p, q}]
DefMetric[{1, 3, 0}, g[-a, -b], CD, PrintAs -> "g", DefInfo -> False]
List @@ Expand[15*SmartSymmetrize[g[-a, -b] g[-c, -d] g[-e, -f]]]
Union@ToCanonicalSym[
  ImposeSym[#, IndexList[-a, -b, -c, -d, -e, -f],
     GenSet[Cycles[{1, 2}], Cycles[{3, 4}], Cycles[{5, 6}]]] & /@ %]
ToCanonical@ExpandSym@%

I hope this helps.

Thomas

[Gaurav Narain]

The index configuration and symmetrisation of indices do not remove duplicate entries. Consider for example the following simple case

g[a,b] g[c,d]

now I want to construct all possible tensors using them which are symmetric in (a,b) and (c,d). when index configuration is applied to this combination it yields 

{g[a,d]g[b,c], g[a,c]g[b,d], g[a,b]g[c,d] }

now when one applies symmetrisation over (a,b) and (c,d) this yields 3 tensors

1) g[a,b]g[c,d]

2) g[a,c]g[b,d]+g[b,c]g[a,d]

3) g[a,d]g[b,c]+ g[b,d]g[a,c]

clear 2 and 3 are same.

well, after posting the original message I released that this issue of repetition can be easily resolved by using the inbuilt command in Mathematica called `DeleteDuplicates’, which is working even for tensors.