Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real $C$-, $G$-structures and sign-coherence of cluster algebras

Ryota Akagi, Zhichao Chen

Abstract

We generalize the theory of integer $C$-, $G$-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to the one of skew-symmetric patterns. In this sense, we extend the sign-coherence of integer $C$-, $G$-matrices proved by Gross-Hacking-Keel-Kontsevich to a more general real class called of quasi-integer type. Furthermore, we give a complete classification of this type by a combinatorial method of real weighted quivers. However, the sign-coherence of real $C$-, $G$-matrices does not always hold in general. For this purpose, we classify all the rank $2$ case and the finite type case via the Coxeter diagrams. We also give two conjectures about the real exchange matrices and $C$-, $G$-matrices. Under these conjectures, the dual mutation, $G$-fan structure and synchronicity property hold. As an application, the isomorphism of several kinds of exchange graphs is studied.

Real $C$-, $G$-structures and sign-coherence of cluster algebras

Abstract

We generalize the theory of integer -, -matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to the one of skew-symmetric patterns. In this sense, we extend the sign-coherence of integer -, -matrices proved by Gross-Hacking-Keel-Kontsevich to a more general real class called of quasi-integer type. Furthermore, we give a complete classification of this type by a combinatorial method of real weighted quivers. However, the sign-coherence of real -, -matrices does not always hold in general. For this purpose, we classify all the rank case and the finite type case via the Coxeter diagrams. We also give two conjectures about the real exchange matrices and -, -matrices. Under these conjectures, the dual mutation, -fan structure and synchronicity property hold. As an application, the isomorphism of several kinds of exchange graphs is studied.

Paper Structure

This paper contains 37 sections, 69 theorems, 148 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Purpose
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. Basic notations
  8. $C$-, $G$-matrices and first duality
  9. Periodicity
  10. Periodicity
  11. Projection of matrix patterns
  12. Quiver setting
  13. Skew-symmetrizing method
  14. Classification of quasi-integer type exchange matrices
  15. Quasi-integer type
  16. ...and 22 more sections

Key Result

Theorem 1.1

For any $t,t' \in \mathbb{T}_n$ and $\sigma \in \mathfrak{S}_n$, the following conditions are equivalent. Moreover, the $G$-fan has the periodicity $\mathcal{C}(G_{t'})=\mathcal{C}(G_{t})$ if and only if the permutation $\sigma \in \mathfrak{S}_{n}$ as above exists.

Figures (15)

  • Figure 1: An $\mathbb{R}$-valued quiver example
  • Figure 2: $C$-pattern of $B=\left(0-\frac{1}{2}20\right)$
  • Figure 3: $G$-pattern of $B=\left(0-\frac{1}{2}20\right)$
  • Figure 4: $G$-pattern and $G$-fan associated with $B=\left(0-\frac{1}{2}20\right)$
  • Figure 5: Type $I_2(7)$
  • ...and 10 more figures

Theorems & Definitions (153)

  • Theorem 1.1: Nak21, Nak23, Synchronicity
  • Theorem 1.2: Theorem \ref{['thm: skew-symmetrizing method']}, Skew-symmetrizing method
  • Theorem 1.3: Theorem \ref{['thm: classification of quasi-integer type']}
  • Theorem 1.4: Theorem \ref{['thm: sign-coherency for quasi-integer matrices']}
  • Theorem 1.5: Theorem \ref{['thm: rank 2 classification']}
  • Theorem 1.6: Theorem \ref{['thm: finite type classifcation']}
  • Conjecture 1.7: Conjecture \ref{['conj: standard hypothesis']}, \ref{['conj: discreteness conjecture']}, \ref{['conj: standard and discreteness conjecture']}
  • Theorem 1.8: Proposition \ref{['prop: dual mutation']}, Theorem \ref{['thm: fan']}
  • Theorem 1.9: Theorem \ref{['thm: equivalency of the periodicity among modified c-, g-vectors']}
  • Theorem 1.10: Theorem \ref{['thm: matrix-cone synchronicity']}, Cone-Matrix Synchronicity
  • ...and 143 more