Mixed symmetry property and degrees of freedom

[sourena]

How can I calculate degrees of freedom of a rank (o,3) tensor, A_abc, that is mixed symmetry and antisymmetric in the first 2 indices? By mixed symmetry I mean this:
A_abc+A_cab+A_bca=0.

 

[MathematicalPhysicist]

You have 3*3*3=27 options for $A_{ijk}$.

Now by antisymmetry terms like A_{abc}=-A_{bac};
A_{cab}=-A_{acb}; A_{bca}=-A_{cba};

So only 27-3=24 terms are independent, now after the mixed symmetry we are left with:
24-1=23 (cause one term depends on the other two).

So we are left with 5 dof.

Hope I helped somehow.
Edit:
Obviously that terms with the same first two indices are zero, and we have 3*2=6 such terms, so we are left with 23-6=17 dof.
After that we have terms like:
A_{abb}=-A_{bab}; A_{acc}=-A_{cac} ; A_{cbb}=-A_{bcb}; A_{caa}=-A_{aca} ; A_{bcc}=-A_{cbc} ; A_{baa}=-A_{aba}
Which means in the end:
17-6=11 dof.

I hope I counted right this time. :-D

 

[MathematicalPhysicist]

BTW this raises a nice programming task of how to compute some arbitrary tensor of rank $0\choose n$ with the above constraints.

Pitty I am not that good programmer.

 

[sourena]

For a rank (0,3) tensor, A_abc, without any constraint, degrees of freedom are 216, a,b,c = 0, ..., 6.

If this tensor is antisymmetric in the first 2 indices, degrees of freedom dicrease to 90.

If it is mixed symmetry, the number of constraint equations are:

$\frac{n(n-1)(n-2)}{3!}$, a,b,c=0, ..., n.

For our example n = 6 so, the number of constraint equations are 20 therefore degrees of freedom are 70.

 

[blue_raver22]

The mixed symmetry property of a tensor refers to the way its components behave under index permutation. In the case of a rank (0,3) tensor, Aabc, being mixed symmetric means that swapping any two indices results in a sign change. In other words, Aabc = -Abac = -Acba. This property is often seen in tensors that describe physical quantities such as stress and strain in materials.

To calculate the degrees of freedom of a mixed symmetric tensor, we first need to understand the concept of degrees of freedom. In general, degrees of freedom refers to the number of independent variables needed to fully describe a physical system. For a tensor, this is equivalent to the number of independent components needed to fully specify the tensor.

For a rank (0,3) tensor, the total number of components is given by the product of the dimensions of each index, which in this case is 3x3x3=27. However, due to the mixed symmetry property, not all of these components are independent. In fact, only 1/3 of the components are independent, as the other 2/3 can be obtained by permuting the indices. This means that the degrees of freedom of a mixed symmetric rank (0,3) tensor is 1/3 of the total number of components, which is 27/3=9.

Now, to calculate the degrees of freedom of a rank (0,3) tensor, Aabc, that is both mixed symmetric and antisymmetric in the first 2 indices, we first need to consider the antisymmetry property. This means that Aabc = -Aacb, which further reduces the number of independent components to 1/2 of the total, or 27/2=13.5. However, since degrees of freedom must be a whole number, we can round this down to 13.

In summary, the degrees of freedom of a rank (0,3) tensor, Aabc, that is mixed symmetric and antisymmetric in the first 2 indices, is 13. This means that there are 13 independent components that are needed to fully specify this tensor.