Geometric complexity theory and matrix powering

Abstract

Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (2001, 2008) [23,24] introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show negative results for the method of shifted partial derivatives (Efremenko et al., 2016 [6]) and for geometric complexity theory (Ikenmeyer and Panova, 2016 [17] and Bürgisser et al., 2016 [3]), in which occurrence obstructions were ruled out to be able to prove superpolynomial complexity lower bounds. Following a classical homogenization result of Nisan (1991) [25] we replace the determinant in geometric complexity theory with the trace of a symbolic matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent.Interestingly—in contrast to the determinant—the trace of a symbolic matrix power is not uniquely determined by its stabilizer.