# The role of symmetry in geometric complexity theory?

I'm not well versed in geometric complexity theory so my question could be trivial. I understand that GCT program studies the symmetries of determinant and permanent to prove Valiant's Hypothesis: $VP\ne VNP$

In simple terms, What is the invariant property? Which mapping (transformation) is applied to the determinant and permanent? How is the symmetry of permanent used to explain the difficulty of computing the permanent?

Second, even characterization by symmetries does not explain hardness - since both $det$ and $perm$ are characterized by their symmetries, but $det$ is easy. The idea instead is this: it should be easier to understand the algebro-geometric and representation-theoretic properties of functions that are characterized by their symmetries, and hence to carry out the GCT plan of attack.
(That characterization by symmetries should make understanding easier can be formalized a tiny bit more as follows. Since GCT is studying the orbits of $det$ and $perm$, and there is a sort of duality between orbits and stabilizers, functions that are characterized by their stabilizers should have "special" orbits in some sense. I guess this alone doesn't actually say anything about how difficult it should be to understand these orbits. But understanding "generic" orbits can be very hard; having a nice property like characterization by symmetries at least gives us something to grab on to that we can try to use to gain more understanding.)