Strange results when combining TensorProduct withTensorWedge

Questions
,
1

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

2

You can use Trace to follow its logic. The 1/4th value comes from the tranformation of this:

a\[TensorWedge]b

into this

(1/2 (a\[TensorProduct]b - TensorTranspose[a\[TensorProduct]b, {2, 1}]))

You’re right that the 1/2 doesn’t seem right . Consider the comparison:

{a, b, c}\[TensorWedge]{x, y, z} // Normal

(1/2 ({a, b, c}\[TensorProduct]{x, y, z} - TensorTranspose[{a, b, c}\[TensorProduct]{x, y, z}, {2, 1}]))
3

Yes, there is a missing multinomial factor in the purely symbolic computation. An unfortunate result of exploring the various possible conventions.

Thank you for reporting this issue!

Jose.