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Synchronicity phenomenon in cluster patterns

Tomoki Nakanishi

Abstract

It has been known that several objects such as cluster variables, coefficients, seeds, and $Y$-seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

Synchronicity phenomenon in cluster patterns

Abstract

It has been known that several objects such as cluster variables, coefficients, seeds, and -seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

Paper Structure

This paper contains 29 sections, 60 theorems, 110 equations.

Table of Contents

  1. Introduction: Synchronicity problem
  2. Synchronicity in cluster patterns of cluster algebras
  3. Cluster patterns
  4. Separation formulas
  5. Further fundamental results
  6. Sign-coherence
  7. Laurent positivity
  8. Synchronicity phenomenon
  9. $\sigma$-periodicity
  10. $xy/GC$ synchronicity
  11. More synchronicity results
  12. Conjectures by Fomin and Zelevinsky
  13. Synchronicity under dualities
  14. Synchronicity for conjugate pair
  15. Open problem
  16. ...and 14 more sections

Key Result

Theorem 3.1

For any $i=1,\dots,n$ and $t\in \mathbb{T}_n$, the function $F_{i;t}(\mathbf{u})$ is a polynomial in $\mathbf{u}$ with coefficients in $\mathbb{Z}$.

Theorems & Definitions (124)

  • Example 2.1
  • Definition 2.2: Seeds
  • Definition 2.3: Seed mutations
  • Definition 2.4: Cluster patterns
  • Example 2.5
  • Theorem 3.1: Fomin07
  • Proposition 3.2: Fomin07
  • Theorem 3.3: Separation Formulas Fomin07
  • Remark 3.4
  • Remark 3.5
  • ...and 114 more