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/* formQ0: Evaluate trace-free decomposition of a symmetric tensor.
*
* Usage: Q0(Q);
*
* parameters: Q: itensor expression
*
* Given the tensor Q that is fully symmetric in all indices, this
* function returns the corresponding STF tensor.
*
* Side effects: None.
*
* Example:
*
* (%i1) load("formQ0.mac")$
* (%i2) load(itensor)$
* (%i3) dim:3$
* (%i4) decsym(g,0,2,[],[sym(all)])$
* (%i5) imetric(g)$
* (%i6) flipflag:true$
* (%i7) defcon(Q,Q,Q)$
* (%i8) decsym(Q,0,3,[],[sym(all)])$
* (%i9) ishow(canform('kdels([a,b,c],[j,k,l])*Q0(Q([],[a,b,c]))))$
* %1 %2 %3 j k l
* 3 Q g kdels
* %1 %2 %3 j k l %1 %2 %3
* (%t9) Q kdels - --------------------------
* %1 %2 %3 5
*
* (c) 2021 Viktor T. Toth (https://www.vttoth.com/)
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License as
* published by the Free Software Foundation; either version 2 of
* the License, or (at your option) any later version.
*
* This program is distributed in the hope that it will be
* useful, but WITHOUT ANY WARRANTY; without even the implied
* warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
* PURPOSE. See the GNU General Public License for more details.
*/
Q0(_Q):=block
([_a,_k,_i,_j,_l,_g],
_a:indices(_Q)[1],
_k:length(_a),
_g:ev(imetric),
_Q+canform(contract(sum(
(-1/2)^_p*(dim+2*_k-2*(_p+2))!!/_p!/(_k-2*_p)!/(dim+2*_k-4)!!
*canform(contract(expand(
kdels(makelist(concat('_i,_j),_j,1,_k),_a)
*prod(_g([],[concat('_i,2*_i-1),
concat('_i,2*_i)])*_g([concat('_j,2*_i-1),concat('_j,2*_i)]),_i,1,_p)
*sublis(makelist(_a[_l]=(flatten([makelist(concat('_j,_i),_i,1,2*_p),
makelist(concat('_i,_i),_i,2*_p+1,_k)]))[_l],_l,1,_k),_Q)))),
_p,1,_k/2)))
)$
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