# finding rank-3 tensors compatible with a rank-2 tensor projection

I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any permutation of the indices, so $\sigma_{ijk}=\sigma_{jik}=\sigma_{ikj}=\sigma_{kij}=\sigma_{jki}=\sigma_{kji}$. The tensor is cubic in size, with indices $i,j,k$ all running from 1 to $N$.

Now, define a rank-2 tensor $\boldsymbol{\Omega}$ (of size $N\times N$) with the following rule $\Omega_{ij}=\sum_k\sigma_{ijk}$. Assume one knows all the values $\Omega_{ij}$ for all $i,j$, but not those of $\sigma_{ijk}$.

Broadly speaking, how much can one know about $\boldsymbol{\sigma}$ on the basis of having complete knowledge of $\boldsymbol{\Omega}$, provided one knows that $\sigma_{ijk}$ can be 0 or 1, as well as the rule $\Omega_{ij}=\sum_k\sigma_{ijk}$? In other words, I think there are multiple tensors $\boldsymbol{\sigma}$ producing the same $\boldsymbol{\Omega}$, but I would like to find a way to narrow those down.

What is the relevant literature on this kind of problem? Does it relate to copulas and Frechet classes?

Thanks

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