Assume an m-way tensor $\mathcal{Z}$.

$\mathcal{Z}_{p_1 p_2 ... p_m} = 0$ if any different indices match and $\mathcal{Z}_{p_1 p_2 ... p_m} = 1$ otherwise.

It is a symmetric tensor. Now if it is 2-way tensor, i.e., a matrix, I can decompose it by diagonalization (of a symmetric hollow matrix). For a general tensor, probably I can do a numerical tensor decomposition (e.g., a symmetric tensor decomposition).

But I was wondering, since it is such a simple tensor (elements are either 1 or 0), is there an analytical decomposed expression for this tensor?

I want to avoid storing the full tensor and then decompose it numerically.

I am a not a mathematician so I apologize if my terminologies are not correct.