Application of weak determinantal identities to GCT?
Determinantal identities can be useful, but perhaps not exactly in the way you think. As far as I know, however, the identities do not all "reduce to" the symmetries of the determinant (except for the fact that the symmetries of the determinant characterize it). Whenever one has some determinantal identities, they can often be massaged into getting equations for the orbit closure of the determinant - the closure of the set of homogeneous polynomials that are linear projections of the determinant. The family of these orbit closures (one for each size $n$) is a natural geometric object capturing the complexity class $\mathsf{VP}_{s}$. 

Some work in this direction by Landsberg &amp; Ressayre was reported here (Section 2.2), but I don't know if the actual paper has appeared yet. 

A somewhat related paper, which gives some equations that vanish on the determinant orbit closure is:


  Landsberg, Manivel, &amp; Ressayre. Hypersurfaces with degenerate duals and the Geometric Complexity Theory program. Comentarii Mathematici Helvetici 88(2):469-484, 2013. (arXiv version) 


That paper, as with Mignon-Ressayre, uses the rank of the Hessian, rather than explicitly from a determinantal identity. However, the rank of the Hessian can be seen as closely related to Segre's identity $\det_{n^2}(Hess(\det_n(A)) = (-1)^{\binom{n+1}{2}}(n-1)\det(A)^{n(n-2)}$, which is the identity reportedly used above.

You might also be interested in the following paper, which shows that any reduction from perm to det which respects the symmetries of perm (see the paper for precise definitions) must have exponential blow-up:


  J. M. Landsberg and Nicolas Ressayre. Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture. Differential Geom. Appl., in press,
   2017. (arXiv version)