Symmetric Tensor Product in Pic. 1 - Explained

In summary, there are three products mentioned in the conversation: the tensor product, the wedge product (also known as the anti-symmetric product), and a third product that represents the symmetric part of a tensor. The name of this third product is not specified.
  • #1
mikeeey
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we all know that a tensor has a symmetric part and anti-symmetric part and the anti-symmetric product (of the anti-symmetric part) called [wedge product] in pic.(2). then what is the name of the product the represents the symmetric part of a tensor in pic.(1) ?
 

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  • #2
If I understood your question correctly, the general product in the exterior algebra of the vector space is the tensor product, which then gives you a graded algebra , where the product of an n-tensor and a k-tensor is an (n+k)-tensor . As you said, the restriction to the alternating tensors uses the wedge product.
 
  • #3
ftp://ftp.cis.upenn.edu/pub/cis610/public_html/diffgeom7.
in this pdf it.s written the three product ( tensor and wedge and the other product ) . well i don't know the name of the third product.
 

1. What is a symmetric tensor product?

A symmetric tensor product is a mathematical operation that combines two tensors to create a new tensor that is symmetric in its indices. This means that the resulting tensor remains unchanged when its indices are permuted.

2. How is the symmetric tensor product denoted?

The symmetric tensor product is denoted by the symbol s, with the subscript s indicating that it is a symmetric operation.

3. What is the difference between the symmetric tensor product and the regular tensor product?

The regular tensor product, denoted by , is a general operation that combines two tensors without any restrictions on the resulting tensor. In contrast, the symmetric tensor product ensures that the resulting tensor is symmetric in its indices.

4. How is the symmetric tensor product calculated?

The symmetric tensor product is calculated by first taking the regular tensor product of the two tensors. Then, the resulting tensor is symmetrized by summing over all possible permutations of its indices and dividing by the number of permutations.

5. What are some applications of the symmetric tensor product?

The symmetric tensor product is commonly used in physics and engineering to describe symmetric systems, such as crystals or molecules. It is also used in algebraic geometry and representation theory to study symmetries of geometric objects and group actions. Additionally, the symmetric tensor product has applications in machine learning and signal processing for feature extraction and dimensionality reduction.

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