Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The periodicity conjecture for pairs of Dynkin diagrams

Bernhard Keller

TL;DR

The paper proves the periodicity conjecture for Y-systems associated with pairs of Dynkin diagrams by reframing the problem in the language of cluster algebras and their 2-Calabi-Yau categorifications. It constructs a 2-CY realization from the triangle product of quivers, uses Zamolodchikov transformations to encode the dynamics, and shows that the induced autoequivalence has finite order dividing $h+h'$, thereby giving a uniform, effective proof of periodicity. The approach also yields an explicit, categorically flavored path to the general solution of the Y-system via homological invariants and quiver Grassmannians. The non simply-laced case is handled via folding, reducing to the simply-laced proof, with the overall result providing a conceptual, and in principle computationally explicit, understanding of Y-system periodicity in this broad setting.

Abstract

We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky's cluster algebras and their (additive) categorification via triangulated categories.

The periodicity conjecture for pairs of Dynkin diagrams

TL;DR

The paper proves the periodicity conjecture for Y-systems associated with pairs of Dynkin diagrams by reframing the problem in the language of cluster algebras and their 2-Calabi-Yau categorifications. It constructs a 2-CY realization from the triangle product of quivers, uses Zamolodchikov transformations to encode the dynamics, and shows that the induced autoequivalence has finite order dividing , thereby giving a uniform, effective proof of periodicity. The approach also yields an explicit, categorically flavored path to the general solution of the Y-system via homological invariants and quiver Grassmannians. The non simply-laced case is handled via folding, reducing to the simply-laced proof, with the overall result providing a conceptual, and in principle computationally explicit, understanding of Y-system periodicity in this broad setting.

Abstract

We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky's cluster algebras and their (additive) categorification via triangulated categories.

Paper Structure

This paper contains 48 sections, 31 theorems, 171 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The periodicity conjecture
  3. On the proof
  4. Contents
  5. The conjecture
  6. Statement
  7. Plan of the proof
  8. Reformulation of the conjecture in terms of cluster combinatorics
  9. Quiver mutation
  10. $Y$-seeds
  11. Products of quivers
  12. Reformulation of the conjecture
  13. More cluster combinatorics
  14. Tropical $Y$-variables and $F$-polynomials
  15. Consequence for the periodicity conjecture
  16. ...and 33 more sections

Key Result

Theorem 2.3

The periodicity conjecture conj:periodicity is true.

Figures (2)

  • Figure 1: The quiver $\vec{A}_4\otimes\vec{D}_5$
  • Figure 2: The quivers $\vec{A}_4\boxtimes\vec{D}_5$ and $\vec{A}_4 \square \vec{D}_5$

Theorems & Definitions (54)

  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.4
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • Proposition 4.2: Proposition 3.12 of FominZelevinsky07
  • Lemma 5.4
  • Theorem 5.7: Amiot Amiot09
  • ...and 44 more