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

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.

💡 Research Summary

The paper is the sixth installment in the series on Geometric Complexity Theory (GCT) and presents a concrete implementation plan for the “flip,” the central methodological principle of GCT. The flip is a two‑step reduction that transforms a lower‑bound problem in complexity theory—such as proving P ≠ NP in characteristic zero—into an upper‑bound (positive) problem about structural constants arising in representation theory and algebraic geometry.

First, the authors observe that many lower‑bound questions can be rephrased as decision problems concerning whether certain representation‑theoretic constants (plethysm coefficients, Kronecker coefficients, Littlewood–Richardson numbers, etc.) are non‑zero. These constants are known to be #P‑hard to compute in general, so a direct algorithmic solution seems impossible. The flip therefore relaxes the original decision problem: instead of asking “Is the constant zero?” one asks “Does there exist a formula for the constant that has only non‑negative integer coefficients?” In other words, the problem is reduced to the existence of a positive expression for the constant.

The second reduction translates this positivity requirement into a purely mathematical hypothesis: the positivity hypothesis. It asserts that each structural constant can be expressed as a sum of monomials with non‑negative integer coefficients, or equivalently that the associated generating functions are saturated and admit a description as a positive integer programming (PIP) instance. Saturation means that if a multiple of a weight yields a non‑zero constant, then the original weight already yields a non‑zero constant—a property familiar from the Knutson–Tao saturation theorem for Littlewood–Richardson coefficients. If saturation holds for plethysm and related constants, the non‑vanishing decision can be encoded as a PIP problem whose constraints and objective function are all non‑negative. Such PIP instances are solvable in polynomial time by interior‑point methods, provided the positivity of the underlying coefficients.

The authors connect the positivity hypothesis to deep results in algebraic combinatorics. The Kazhdan–Lusztig (KL) polynomials, which govern the change‑of‑basis coefficients between standard and canonical bases in Hecke algebras, are known to have non‑negative coefficients. Their positivity proofs rely on the Deligne–Weil theory of étale cohomology and, crucially, on the Riemann hypothesis over finite fields (RHFF). Similarly, the structure constants of the global crystal (canonical) bases of Drinfeld–Jimbo quantum groups are positive, and their proofs again invoke RHFF. Thus, establishing the positivity hypothesis for the GCT constants would follow from these existing positivity theorems, provided the same cohomological machinery applies.

The paper outlines a concrete roadmap:

1. Saturation Proofs – Extend the known saturation results (e.g., for Littlewood–Richardson coefficients) to plethysm and Kronecker coefficients. This may involve new geometric invariant theory (GIT) arguments or combinatorial models.
2. Positive Integer Programming Formulation – Translate each saturated constant into a PIP instance. Show that the feasible region is described by a system of linear inequalities with non‑negative coefficients, and that the objective function corresponds to the constant itself.
3. Positivity via KL and Canonical Bases – Demonstrate that the coefficients appearing in the PIP formulation are precisely KL polynomials or canonical‑basis structure constants. Invoke the known RHFF‑based positivity proofs to guarantee non‑negativity.
4. Algorithmic Consequence – Apply polynomial‑time interior‑point algorithms to solve the PIP instances, thereby deciding non‑vanishing of the original constants in deterministic polynomial time.

If all steps succeed, the flip is completed: the original lower‑bound problem (e.g., P ≠ NP) is reduced to a polynomial‑time algorithm for a family of decision problems about representation‑theoretic constants. The authors emphasize that the entire chain hinges on the truth of the Riemann hypothesis over finite fields; if RHFF were disproved, the positivity arguments would collapse, and the flip would not yield the desired complexity‑theoretic consequences.

In summary, the paper proposes that the P vs. NP question in characteristic zero can be “flipped” into a question about the existence of positive, saturated formulas for certain structural constants. By leveraging deep positivity results that depend on RHFF, the authors argue that these decision problems belong to the class P, thereby providing a novel bridge between computational complexity, algebraic geometry, representation theory, and arithmetic geometry. The work thus positions the resolution of one of the most famous problems in computer science as intimately linked to some of the most profound conjectures in modern mathematics.