Symmetric Tensor Product in Pic. 1 - Explained

[mikeeey]

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) ?

 

[WWGD]

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.

 

[mikeeey]

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.