Evaluating Symmetric/Antisymmetric Tensors Rank 5 Dimension 999

[Rahat]

please if possible send me the evaluation steps of independent components of a symmetric and antisymmetric tensor of rank 5 and dimension 999.

 

[robphy]

Sounds like a homework problem.
Start with something simple first... like rank 2 with dimension 3.

 

[tacman]

Evaluating symmetric/antisymmetric tensors of rank 5 and dimension 999 involves determining the independent components of the tensor. This can be done by following a few key steps.

Step 1: Understand the properties of symmetric and antisymmetric tensors
Symmetric tensors have the property that their components are unchanged under any permutation of indices. This means that the value of the tensor remains the same even if the order of the indices is changed. On the other hand, antisymmetric tensors have the property that their components change sign under any permutation of indices. This means that the value of the tensor changes sign when the order of the indices is changed.

Step 2: Determine the total number of independent components
For a symmetric tensor of rank 5 and dimension 999, the total number of independent components can be calculated using the formula:
n = (r+d-1)!/(r!(d-1)!)
where n is the number of independent components, r is the rank of the tensor, and d is the dimension of the tensor. In this case, n = (5+999-1)!/(5!(999-1)!) = 10,010,004,999,995.

For an antisymmetric tensor of rank 5 and dimension 999, the total number of independent components can be calculated using the formula:
n = (r+d)!/(r!d!)
where n is the number of independent components, r is the rank of the tensor, and d is the dimension of the tensor. In this case, n = (5+999)!/(5!999!) = 1,998,000,000.

Step 3: Identify the independent components
To identify the independent components of the tensor, we can use the concept of Young tableaux. This involves arranging the indices of the tensor in a specific way to identify the independent components.

For a symmetric tensor of rank 5, the indices can be arranged in a symmetric Young tableau with 5 rows and 5 columns. The number of boxes in each row represents the number of indices in that row, and the number of boxes in each column represents the number of indices in that column. The independent components can then be identified by considering the boxes in the tableau that are not repeated in any other row or column.

For an antisymmetric tensor of rank 5, the indices can be arranged in an antisymmetric Young tableau with 5 rows and 5 columns. The independent components can then