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The cluster complex for cluster Poisson varieties and representations of acyclic quivers

Carolina Melo, Alfredo Nájera Chávez

Abstract

Let $\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $Δ^+(\mathcal{X})$ was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on $\mathcal{X}$ that restrict to a character of a seed torus. Every seed ${ \bf s}$ for $\mathcal{X}$ determines a fan realization $Δ^+_{\bf s}(\mathcal{X})$ of $Δ^+(\mathcal{X})$. For every ${\bf s}$ we provide a simple and explicit description of the cones of $Δ^+_{\bf s}(\mathcal{X})$ and their facets using ${\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $Δ^+_{ \bf s}(\mathcal{X})$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $Δ^+_{\bf s}(\mathcal{X})$ using ${\bf g}$-vectors of (non-necessarily rigid) objects in $\mathsf{K}^{\rm b}(\text{proj} \; kQ)$.

The cluster complex for cluster Poisson varieties and representations of acyclic quivers

Abstract

Let be a skew-symmetrizable cluster Poisson variety. The cluster complex was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on that restrict to a character of a seed torus. Every seed for determines a fan realization of . For every we provide a simple and explicit description of the cones of and their facets using -vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of in terms of -polynomials. In case is skew-symmetric and the quiver associated to is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of using -vectors of (non-necessarily rigid) objects in .
Paper Structure (20 sections, 10 theorems, 76 equations, 5 figures, 3 tables)

This paper contains 20 sections, 10 theorems, 76 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. The skew-symmetrizable case
  4. The skew-symmetric acyclic case
  5. Organization
  6. Background on cluster varieties
  7. Cluster varieties
  8. The weight map
  9. Description of the cluster complex for cluster Poisson varieties
  10. Description of the cones and their supporting hyperplanes via c-vectors
  11. Dimension and facets of the cones
  12. The skew-symmetrizable case
  13. Global monomials on cluster Poisson varieties
  14. Normal vectors via g-vectors in the acyclic case
  15. The cluster ensemble map via representation theory
  16. ...and 5 more sections

Key Result

Theorem 1

$(1)$ For every $\textbf{s}'\in \mathbb T_n$ the cone $(p^*)^{-1}(\mathcal{G}_{\textbf{s}'})$ is a cone of the fan $\underline{\Delta^+_{\textbf{s}}(\mathcal{X} )}$ and each of its cones arises in this way. Moreover, if for $\beta\in N_{\mathbb{R} }$ we let $\beta_\textbf{s}$ be the column vector ob $(2)$ If $p^*(\beta) \in \mathcal{G}_{\textbf{s}'}$ then the theta function parametrized by $\beta_

Figures (5)

  • Figure 1: Supporting hyperplanes in the Auslander-Reiten quiver in $\text{mod } kQ$.
  • Figure 2: Supporting hyperplanes of $\underline{\Delta_{\textbf{s}}^+(\mathcal{X} )}$.
  • Figure 3: Cluster complexes $\underline{\Delta_{\textbf{s}}^+(\mathcal{A} )}$ and $\underline{\Delta_{\textbf{s}}^+(\mathcal{X} )}$.
  • Figure 4: $\underline{\Delta_{\textbf{s}}^+(\mathcal{X} )}$.
  • Figure 5: The fan in $N_{\mathbb{R} }/\text{ker}(p^*)$ induced by $\underline{\Delta^+_{\textbf{s}}(\mathcal{X} )}$.

Theorems & Definitions (32)

  • Theorem
  • Theorem : Theorem \ref{['dv-gv']}
  • Remark 2.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • ...and 22 more