Geometric Complexity Theory:

>**Theorem (Mulmuley and Sohoni [\[MS\]][1])**
The permanent (respectively the determinant) polynomial is characterized by its symmetry group. 


That is if $P$ is a homogeneous polynomial of degree $m$ in $m^2$ variables and its symmetry group $G_P$ also fixes the permanent (respectively the determinant), then $P$ must be a scalar multiple of the permanent (respectively the determinant). 

Landsberg and Ressayre [\[LR\]][2] made progress on Valiant's version of P vs NP using this result.

  [1]: http://epubs.siam.org/doi/pdf/10.1137/S009753970038715X
  [2]: http://%20https://arxiv.org/pdf/1508.05788v2.pdf