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Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

Fang Li, Jie Pan

TL;DR

This work addresses constructing a recurrence formula for cluster variables via $g$-vectors and building a polytope-based, universally positive basis for upper cluster algebras using Newton polytopes of $F$-polynomials in the TSSS setting.By introducing polytopes $N_h$ and polytope functions $\rho_h$, the authors generalize $F$-polynomials and cluster variables, establishing key properties such as indecomposability, positivity, and mutation compatibility through $L^t$-maps and essential skeletons.The main achievement is the polytope basis $\mathcal{P}=\{\rho_h\}$, which forms a strongly positive ${\mathbb Z}{\mathbb P}$-basis for the upper cluster algebra with principal coefficients (and for the intermediate algebra in general semifields), alongside explicit links among $F$-polynomials, $g$-vectors, $d$-vectors, and cluster variables, including proofs of positivity and Fei-type conjectures.The framework provides a constructive, rank-aware approach (starting from rank-2 explicit polytopes) that extends to arbitrary rank, and supplies tools for analyzing mutation effects, polytope faces, and projections, with potential impact on positivity results and combinatorial interpretations in cluster theory.

Abstract

In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$ consisting of polytope functions $ρ_{h}$ corresponding to $N_{h}$. The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector; (ii) proving the subset $\mathcal{P}$ of $\widehat{\mathcal{P}}$ consisting of Laurent polynomials in $\widehat{\mathcal{P}}$ is a strongly positive $\Z Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal A$ over arbitrary semifield $\mathbb P$ in general, $\mathcal{P}$ is a strongly positive $\Z¶$-basis for the intermediate cluster subalgebra $\mathcal{I_P(A)}$ of $\mathcal{U(A)}$. We call $\mathcal P$ the {\em polytope basis}; (iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively.

Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

TL;DR

This work addresses constructing a recurrence formula for cluster variables via $g$-vectors and building a polytope-based, universally positive basis for upper cluster algebras using Newton polytopes of $F$-polynomials in the TSSS setting.By introducing polytopes $N_h$ and polytope functions $\rho_h$, the authors generalize $F$-polynomials and cluster variables, establishing key properties such as indecomposability, positivity, and mutation compatibility through $L^t$-maps and essential skeletons.The main achievement is the polytope basis $\mathcal{P}=\{\rho_h\}$, which forms a strongly positive ${\mathbb Z}{\mathbb P}$-basis for the upper cluster algebra with principal coefficients (and for the intermediate algebra in general semifields), alongside explicit links among $F$-polynomials, $g$-vectors, $d$-vectors, and cluster variables, including proofs of positivity and Fei-type conjectures.The framework provides a constructive, rank-aware approach (starting from rank-2 explicit polytopes) that extends to arbitrary rank, and supplies tools for analyzing mutation effects, polytope faces, and projections, with potential impact on positivity results and combinatorial interpretations in cluster theory.

Abstract

In this paper, we study the Newton polytopes of -polynomials in a TSSS cluster algebra and generalize them to a larger set consisting of polytopes associated to vectors as well as consisting of polytope functions corresponding to . The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its -vector; (ii) proving the subset of consisting of Laurent polynomials in is a strongly positive -basis for consisting of certain universally indecomposable Laurent polynomials when is a cluster algebra with principal coefficients. For a cluster algebra over arbitrary semifield in general, is a strongly positive -basis for the intermediate cluster subalgebra of . We call the {\em polytope basis}; (iii) constructing some explicit maps among corresponding -polynomials, -vectors, -vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively.
Paper Structure (19 sections, 42 theorems, 155 equations, 6 figures)

This paper contains 19 sections, 42 theorems, 155 equations, 6 figures.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. Notions and notations on cluster algebras and Laurent polynomials
  4. Notions and notations on polytopes
  5. Main contents
  6. Some needful conclusions
  7. Polytope associated to an integer vector and the relevant polytope functions
  8. Outline of the idea of the polytope function $\rho_h$ associated to $h\in{\mathbb Z}^n$
  9. Polytope $N_h$ and polytope function $\rho_h$ in rank 2 case
  10. Construction program of $N_h$ and $\rho_h$
  11. Main theorem on $N_h$ and $\rho_h$ in any rank case
  12. Special case: polytopes associated to cluster variables
  13. Recurrence formula and positivity of a cluster variable
  14. On a conjecture of Fei
  15. Relations among $F$-polynomials, $d$-vectors, $g$-vectors and cluster variables
  16. ...and 4 more sections

Key Result

Theorem 1.8

FZ4For any cluster algebra ${\mathcal{A}}$ and any vertices $t$ and $t{^{\prime}}$ in ${\mathbb T}_{n}$, the cluster variable $x_{l;t}$ can be expressed as where

Figures (6)

  • Figure 2: The polytope $N_{h}$ can be divided into three areas.
  • Figure 4: The shape of $N_{h}$ for $h\in{\mathbb Z}^{2}$ up to reflections.
  • Figure 6: The mutation of $N_{h}$ in direction $k$.
  • Figure 7: The polytope $N_{h}$ and its intersections with hyperplanes $z_{3}=i$ for $i\in[0,3]$.
  • Figure 8: The Newton polytope $N_{3;t}$
  • ...and 1 more figures

Theorems & Definitions (83)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 73 more