Geometric Complexity Theory:
Theorem (Mulmuley and Sohoni [MS]) 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] made progress on Valiant's version of P vs NP using this result.