Introduction to Geometric Complexity Theory II
summary
Program reframes permanent-versus-determinant as whether the padded permanent lies in the orbit closure of det under GL_{n^2+1} and uses algebraic geometry plus representation theory to turn that into multiplicity checks. The concrete obstruction sought is an irreducible GL-module with multiplicity zero on the determinant orbit but positive for the permanent, which reduces to vanishing of specific Kronecker coefficients (and plethysm-type multiplicities). Orbits of det under SL are closed, padded permanent orbit is not, and no definitive lower bound emerges yet.
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