EDIT: I thought on rephrasing the question in another way:


I have been working recently with a tensor that satisfies 


$A\_{ijkl}=A\_{i+b,j+b,k+b,l+b}$ $\forall$ $i,j,k,l$ $\in$ Z 


$$dist(i,j,k,l)\leq M$$


where all indices are meant to be integers (also b with $b\geq 0$), and dis(i,j,k,l) is the distance between all pair of indices, so $|i-j|\leq M$, $|i-k|\leq M$ etc... with 6 total distances. Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$ and generate the other elements from the symmetry relation above. 


I am struggling to see this, as for example, considering the case of a matrix $C\_{i+b,j+b}=C\_{i,j}$ it is clear to me how can one do this: one just calculates values for a single row, say $i=0$ and since $C\_{ij}=0\forall |i-j|>M$, one is left with $2M+1$ independent terms. Then, the rest of matrix elements can be derived by using the symmetry relation:


$$C\_{i+b,j+b}=C\_{i,j}$$


However, for the $A$ tensor above I am having difficulties to see this and how one could in principle recover all the missing elements of the set, if I set $i=0$ and calculate for the other indices. How can one get, for example $A\_{1,1,2,3}$ if we only now those terms for $i=0$ ( that is, we know $A\_{0,jkl}$ only ) for the case $b=1$? 


Thanks !!



Symmetric tensor components

> 
> Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$ and generate the other elements from the symmetry relation above.
> 
> 
> 


The issue is very simple. For each integer $i’$ there exists $i $ defined above such that $i’=i+tb$ for some integer $t$. Then 
$$A\_{i’j’k’l’}=A\_{i+tb,(j’-tb)+tb, (k’-tb)+tb,(l’-tb)+tb}= A\_{i,j’-tb, k’-tb,l’-tb}.$$


Remark that the addition of $tb$ to each index keeps the distances between the indices. 



> 
> for example $A\_{1,1,2,3}$ if we only now those terms for $i=0$ ( that is, we know $A\_{0,jkl}$ only ) for the case $b=1$?
> 
> 
> 


$A\_{1,1,2,3}=A\_{0+b,0+b,1+b,2+b}= A\_{0,0,1,2}$.


