Could someone please help me out with this problem? I would like to know what the number of distinct isotropic tensors of rank N is. I am new to the field and got confused by apparently contradictory information I have found.
I thought that for rank 0, any scalar is an isotropic tensor. For rank 1, there is none. For rank 2, Kronecker delta should be the only one and for rank 3 Levi-Civita symbol should also be the only one. From then on, I got lost. According to http://mathworld.wolfram.com/IsotropicTensor.html, there should be 3 for rank 4 and 6 for rank 5. But in a recent lecture I went to it was mentioned that any higher order isotropic tensor can be constructed as a linear combination of Kronecker deltas and Levi-Civita symbols. Therefore my question is whether the number of isotropic tensors in Wolfram Mathworld represents something like a size of the base from which I construct the tensors or whether it is a real number of distict tensors. Also, how do I prove that Kronecker delta is the only isotropic tensor of rank 2 (and Levi-Civita of 3)?
Thanks a lot for answers.
SF
Best Answer
The problem has been solved and appears here:
http://nvlpubs.nist.gov/nistpubs/jres/79b/jresv79bn1-2p49_a1b.pdf