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

TL;DR

This paper discusses a plan within geometric complexity theory to reduce the P vs. NP problem to showing certain structural constants in algebraic geometry and representation theory are positive, linking it to deep conjectures like the Riemann hypothesis.

Contribution

It proposes a method to implement the flip in GCT by reducing complexity problems to positivity hypotheses involving structural constants and their relations to Kazhdan-Lusztig polynomials.

Findings

Positivity of structural constants relates to the Riemann hypothesis over finite fields.

Reduction of complexity problems to positivity hypotheses.

Framework for showing decision problems belong to P.

Abstract

This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In essence, it reduces the negative hypothesis in complexity theory (the lower bound problems), such as the P vs. NP problem in characteristic zero, to the positive hypothesis in complexity theory (the upper bound problems): specifically, to showing that the problems of deciding nonvanishing of the fundamental structural constants in representation theory and algebraic geometry, such as the well known plethysm constants--or rather certain relaxed forms of these decision probelms--belong to the complexity class P. In this article, we suggest a plan for implementing the flip, i.e., for showing that these relaxed decision problems belong to P. This is based on the reduction of the preceding complexity-theoretic positive hypotheses to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae--i.e. formulae with nonnegative coefficients--for the structural constants under consideration and certain functions associated with them. These turn out be intimately related to the similar positivity properties of the Kazhdan-Lusztig polynomials and the multiplicative structural constants of the canonical (global crystal) bases in the theory of Drinfeld-Jimbo quantum groups. The known proofs of these positivity properties depend on the Riemann hypothesis 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 Riemann hypothesis over finite fields and related problems.