Explicit Lower Bounds via Geometric Complexity Theory

摘要

We prove the lower bound R(M_m) ≥ 3/2m~2 - 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense 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 an explicit description of the highest weight vectors in Sym~d(directX)~3 (C~n)~* in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and Schur-Weyl duality.