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Exchange graphs of cluster algebras have the non-leaving-face property

Changjian Fu, Shengfei Geng, Pin Liu

Abstract

The claim in the title is proved.

Exchange graphs of cluster algebras have the non-leaving-face property

Abstract

The claim in the title is proved.
Paper Structure (5 sections, 8 theorems, 12 equations)

This paper contains 5 sections, 8 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Background on cluster algebras
  3. Non-Leaving-Face property of exchange graphs
  4. Background on exchange graphs and non-leaving-face property
  5. Main result

Key Result

Theorem 2.1

For each vertex $t$ in $\mathbb{T}_n$, the matrix $C_t^{B_s;s}$ is inveritble over $\mathbb{Z}$ and each column vector is sign-coherent.

Theorems & Definitions (18)

  • Theorem 2.1: sign-coherence of $c$-vectors
  • Proposition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6: CGY2021
  • Theorem 3.7: existence and uniqueness of Bongartz completion
  • Lemma 3.8
  • ...and 8 more