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I have functions with some symmetry properties, how I have to impose the corresponding conditions within Mathematica, in order to perform all simplifications automatically? f4 - fully antisymmetric, while f2 satisfies the commutation relation:

f2[i2p,j1]+f2[j1,i2p]==KroneckerDelta[j1,i2p];

Also each term in the following expressions has to change its sign under permutation i1 with i2 or j1 with j2. For example, we have

f2[i1,j2]f2[i1p,i2]f2[i2p,j1]-f2[i1p,i1]f2[i2,j2]f2[i2p,j1]
-f2[i1,j1] f2[i1p, i2] f2[i2p, j2] +f2[i1p, i1] f2[i2, j1] f2[i2p, j2] == 
  4 f2[i1, j2] f2[i1p, i2] f2[i2p, j1];

and

f2[i2p, j2]f4[i1p,i1,i2,j1] -f2[j2,i2p]f4[i1,i1p,i2,j1]
==(f2[i2p,j2]+f2[j2,i2p])f4[i1p, i1, i2, j1];

f2[i2,i1p]f4[i2p,i1,j2,j1]-f2[i1,i1p]f4[i2p,i2,j2,j1]
==2*f2[i2, i1p] f4[i2p, i1, j2, j1];
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  • $\begingroup$ The definition of an antisymmetric function of two variable is f [x_,y_]/;Not[OrderedQ[{x,y}]] :=−f [y,x]. Taken from Grozin pag. 39 $\endgroup$
    – vi pa
    Jan 29, 2020 at 20:39
  • $\begingroup$ The following upvalue seem to work f2[x_,y_]+f2[y_,x_]^:=KroneckerDelta[x,y] $\endgroup$
    – vi pa
    Jan 29, 2020 at 21:35

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