Hello,
I'm trying to figure out the symbolic tensors in Mathematica, but I'm getting some odd results. I have 4 vectors (a, b, c, d)
$Assumptions =
(a | b | c | d) \[Element] Arrays[{3}, Reals];
and I want to calculate a tensor product
(a\[TensorWedge]b)\[TensorProduct](c\[TensorWedge]d)
The output of this is
1/4 (a\[TensorProduct]b -
TensorTranspose[
a\[TensorProduct]b, {2, 1}])\[TensorProduct](c\[TensorProduct]d -
TensorTranspose[c\[TensorProduct]d, {2, 1}])
My question is why is that factor of 1/4 there? According to the manual when TensorWedge[a,b] is equal to Multinomial[TensorRank, TensorRank]*
Symmetrize[TensorProduct[a,b], Antisymmetric]. If I substitute this expression in the tensor product, i.e. calculate
(Multinomial[TensorRank[a], TensorRank[b]]*
Symmetrize[a\[TensorProduct]b,
Antisymmetric[All]])\[TensorProduct](Multinomial[TensorRank[c],
TensorRank[d]]*Symmetrize[c\[TensorProduct]d, Antisymmetric[All]])
I get
(a\[TensorProduct]b -
TensorTranspose[
a\[TensorProduct]b, {2, 1}])\[TensorProduct](c\[TensorProduct]d -
TensorTranspose[c\[TensorProduct]d, {2, 1}])
Which is the same as before, but without the 1/4 coefficient. However, if I use numerical tensors and calculate explicitly TensorWedge[a,b] and TensorWedge[b,c] in separate cells and paste the results into
(a\[TensorWedge]b)\[TensorProduct](c\[TensorWedge]d)
I also get the result without the 1/4 coefficient. It seems to me that Mathematica forgets about the Multinomial coefficients when doing the tensor product with symbolic tensors. Does anyone know why this is happening? I haven't found anything in the manual that would explain this behaviour. It seems like a bug to me.
Greetings,
Richard
