Slice rank and partition rank of the determinant

The Laplace expansion expresses the $n \times n$ determinant $\det_n$ as a sum of $n$ products. Do shorter expansions exist? In this paper we: - Fully determine the slice rank decompositions of $\det_n$ (where each product must contain a linear factor): In this case, we show that $n$ summands are necessary, and moreover, the only such expansions with $n$ summands are equivalent (in a precise sense) to the Laplace expansion. - Prove a logarithmic lower bound for the partition rank of $\det_n$ (where each product is of multilinear forms): In this case, we show that at least $\log_2(n)+1$ summands are needed. We also explain why existing techniques fail to yield any nontrivial lower bound, and why our new method cannot give a super-logarithmic lower bound. - Separate partition rank from slice rank for $\det_n$: we find a quadratic expansion for $\det_4$, over any field, with fewer summands than the Laplace expansion. This construction is related to a well-known example of Green-Tao and Lovett-Meshulam-Samorodnitsky disproving the naive version of the Gowers Inverse conjecture over small fields. An important motivation for these questions comes from the challenge of separating structure and randomness for tensors. On the one hand, we show that the random construction fails to separate: for a random tensor of partition rank $r$, the analytic rank is $r-o(1)$ with high probability. On the other hand, our results imply that the determinant yields the first asymptotic separation between partition rank and analytic rank of $d$-tensors, with their ratio tending to infinity with $d$.