I have recently managed to convince myself that I know what tensors are. I convinced myself by first studying rotations and then learning about transformation rules and such.
A question that has been asked and answered many times is that of finding the number of independent components of a tensor. I am asking it again here in hopes of finally understanding how to do it.
The question I am looking at goes something like
Given $R_{iklm}$ (i) ---> 256
And the conditions
$R_{iklm} = -R_{ikml} = -R_{kilm}$ (ii) ---> reduces to 36
$R_{iklm} = -R_{lmik} $ (iii) ---> reduces to 21
$R_{iklm} + R_{ilmk} + R_{imkl} = 0$ (iv)---> reduces to 20
I happen to believe that the 256 is from 4 times 4 times 4 times 4.
What I did was construct $R_{iklm}$ as a tensor product of two rank 2 tensors.
There are many ways of breaking it up but the diagonals are seen to be zeros.
So I counted out 16 * 4 diagonal components = 64
I also counted out the repeating lower diagonal stuff = 6 * 16 = 96 components
Then I counted out the diagonals of the upper diagonal 6 *4 = 24
I also counted out the lower diagonals of the 6 * 6 = 36
The tally is 256 - 64 -96 -24 -36 = 36
Hypercube diagonal planes vanish.
We are left with 36. This should take one from (i) to (ii)
Here is a picture I made with wolfram that probably gives away the reasoning.
How to go from (ii) to (iii)?
and (iii) to (iv)?