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This article belongs to a series on geometric complexity theory (GCT), anapproach to the P vs. NP and related problems through algebraic geometry andrepresentation theory. The basic principle behind this approach is called theflip. In essence, it reduces the negative hypothesis in complexity theory (thelower bound problems), such as the P vs. NP problem in characteristic zero, tothe positive hypothesis in complexity theory (the upper bound problems):specifically, to showing that the problems of deciding nonvanishing of thefundamental structural constants in representation theory and algebraicgeometry, such as the well known plethysm constants--or rather certain relaxedforms of these decision probelms--belong to the complexity class P. In thisarticle, we suggest a plan for implementing the flip, i.e., for showing thatthese relaxed decision problems belong to P. This is based on the reduction ofthe preceding complexity-theoretic positive hypotheses to mathematicalpositivity hypotheses: specifically, to showing that there exist positiveformulae--i.e. formulae with nonnegative coefficients--for the structuralconstants under consideration and certain functions associated with them. Theseturn out be intimately related to the similar positivity properties of theKazhdan-Lusztig polynomials and the multiplicative structural constants of thecanonical (global crystal) bases in the theory of Drinfeld-Jimbo quantumgroups. The known proofs of these positivity properties depend on the Riemannhypothesis over finite fields and the related results. Thus the reduction here,in conjunction with the flip, in essence, says that the validity of the P vs.NP conjecture in characteristic zero is intimately linked to the Riemannhypothesis over finite fields and related problems.

Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry