An array $\mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that $$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$ for all values of $i,j,k,l$. The number of independent components in this array is
I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.
Sorry for my bad english, this is my first question.
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices $$ \{i,j\} =\{1,1\}, \{1,2\}, \{2,2\}, \{1,3\},\ldots\{4,4\} $$ in total $k={n\choose 2}+n={4\choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10\times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each $\{i,j\}=\{k,l\}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
