Return to Question

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

where all indices are meant to be integers (also b with $b\geq 0$). Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$. Let's say that the $supp(A)=(2N+1)^{4}$ with $|i|,|j|,|k|,|l|\leq N$, where $supp(A)$ represents the support of the tensor. Then, it is argued that the total number of independent elements is given by:

$ (b-1)(2N+1)^{3}$

as, according to the first equation, the rest of elements can be derived by the symmetry. 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. 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 (the $(2N+1)^{4} - (b-1)(2N+1)^{3}$ remaining elements )

Thanks !!

EDIT:

I forgot to add that the tensor must satisfy dist(i,j,k,l)≤N. This is also confusing me, as what is meant here by dist(i,j,k,l) is meant by the pair combinations of indices, so |i−j|≤N , |i−k|≤N. |i−l|≤N ... and so on. Also, the matrix is required to satisfy the symmetry above with |i−j|≤N. In that case, you can fix one of the matrix indices i and compute a single row; the rest follows by symmery but this is what I don't see in the tensor case

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 !!

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

where all indices are meant to be integers (also b with $b\geq 0$). Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$. Let's say that the $supp(A)=(2N+1)^{4}$ with $|i|,|j|,|k|,|l|\leq N$, where $supp(A)$ represents the support of the tensor. Then, it is argued that the total number of independent elements is given by:

$ (b-1)(2N+1)^{3}$

as, according to the first equation, the rest of elements can be derived by the symmetry. 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. 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 (the $(2N+1)^{4} - (b-1)(2N+1)^{3}$ remaining elements )

Thanks !!

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

where all indices are meant to be integers (also b with $b\geq 0$). Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$. Let's say that the $supp(A)=(2N+1)^{4}$ with $|i|,|j|,|k|,|l|\leq N$, where $supp(A)$ represents the support of the tensor. Then, it is argued that the total number of independent elements is given by:

$ (b-1)(2N+1)^{3}$

as, according to the first equation, the rest of elements can be derived by the symmetry. 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. 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 (the $(2N+1)^{4} - (b-1)(2N+1)^{3}$ remaining elements )

Thanks !!

EDIT:

I forgot to add that the tensor must satisfy dist(i,j,k,l)≤N. This is also confusing me, as what is meant here by dist(i,j,k,l) is meant by the pair combinations of indices, so |i−j|≤N , |i−k|≤N. |i−l|≤N ... and so on. Also, the matrix is required to satisfy the symmetry above with |i−j|≤N. In that case, you can fix one of the matrix indices i and compute a single row; the rest follows by symmery but this is what I don't see in the tensor case