The determinant has slice rank n, partition rank at least log2(n)+1, and the 4x4 determinant has partition rank 3, giving the first unbounded separation between partition rank and analytic rank.
Derksen, On the nuclear norm and the singular value decomposition of tensors, Found
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Slice rank and partition rank of the determinant
The determinant has slice rank n, partition rank at least log2(n)+1, and the 4x4 determinant has partition rank 3, giving the first unbounded separation between partition rank and analytic rank.