Geometric complexity theory and tensor rank

Mulmuley and Sohoni GCT1, SICOMP 2001; GCT2, SICOMP 2008 proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G =GL(W1) x GL(W2) x GL(W3) acting on the tensor product W=W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs =SL(W1) x SL(W2) x SL(W3). A key idea from GCT2 is that the irreducible Gs-representations occurring in the coordinate ring of the G-orbit closure of a stable tensor w ∈ W are exactly those having a nonzero invariant with respect to the stabilizer group of w.