Orbit classification of determinant-tensor decompositions

Investigate the orbit of tensor rank decompositions of the \(n \times n\) determinant tensor \(\operatorname{det}_n\) under the diagonal action of \(\operatorname{SL}_n\) for general \(n\), including whether known decompositions belong to the same orbit.

Background

The $n \times n$ determinant tensor is invariant under the diagonal action of $\operatorname{SL}_n$, so applying this group action to any tensor rank decomposition produces another decomposition of the same tensor. The paper verifies that two known decompositions lie in the same $\operatorname{SL}_n$-orbit for $n=3$ and $n=4$. A general description or classification of these orbits for arbitrary $n$ remains unresolved.