  We prove the lower bound R(M_m) \geq 3/2 m^2 - 2 on the border rank of m x m
matrix multiplication by exhibiting explicit representation theoretic
(occurence) obstructions in the sense of the geometric complexity theory (GCT)
program. While this bound is weaker than the one recently obtained by Landsberg
and Ottaviani, these are the first significant lower bounds obtained within the
GCT program. Behind the proof is the new combinatorial concept of obstruction
designs, which encode highest weight vectors in Sym^d\otimes^3(C^n)^* and
provide new insights into Kronecker coefficients.
