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Newton polytopes in cluster algebras and $τ$-tilting theory

Peigen Cao

TL;DR

The paper addresses whether Newton polytopes of $F$-polynomials determine key objects in cluster algebras and $\tau$-tilting theory. It develops the $F$-invariant and its module-analogue to connect polytope geometry with algebraic structure, using left Bongartz completion and reduction arguments. The main results show that cluster monomials in non-initial variables are uniquely determined by $P(F_u^{t_0})$ and that indecomposable $\tau$-rigid modules and left finite multi-semibricks are likewise determined by the Newton polytopes of their $F$-polynomials, i.e., by $P(F_M)$ or $\mathsf P(M)$. These findings link cluster algebras and $\tau$-tilting theory through Newton polytopes, offering a polyhedral lens on classification and mutation phenomena with potential implications for parametrization by polytopes in representation theory and categorification.

Abstract

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $τ$-rigid modules and the left finite multi-simibricks in $τ$-tilting theory are uniquely determined by their Newton polytopes of these modules.

Newton polytopes in cluster algebras and $τ$-tilting theory

TL;DR

The paper addresses whether Newton polytopes of -polynomials determine key objects in cluster algebras and -tilting theory. It develops the -invariant and its module-analogue to connect polytope geometry with algebraic structure, using left Bongartz completion and reduction arguments. The main results show that cluster monomials in non-initial variables are uniquely determined by and that indecomposable -rigid modules and left finite multi-semibricks are likewise determined by the Newton polytopes of their -polynomials, i.e., by or . These findings link cluster algebras and -tilting theory through Newton polytopes, offering a polyhedral lens on classification and mutation phenomena with potential implications for parametrization by polytopes in representation theory and categorification.

Abstract

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their -polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the -rigid modules and the left finite multi-simibricks in -tilting theory are uniquely determined by their Newton polytopes of these modules.
Paper Structure (16 sections, 47 theorems, 100 equations, 1 table)

This paper contains 16 sections, 47 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on polytopes and cluster algebras
  3. Polytopes, Minkowski sum and tropical polynomials
  4. Cluster algebras
  5. Bongartz completion in cluster algebras
  6. $\tau$-tilting theory
  7. Decorated modules and $\tau$-tilting pairs
  8. $g$-vectors and $F$-polynomials of decorated modules
  9. Bongartz completion in $\tau$-tilting theory
  10. $F$-invariant and Newton polytopes of $F$-polynomials
  11. $F$-invariant of cluster monomials
  12. Newton polytopes of $F$-polynomials
  13. $F$-invariant and Newton polytopes of $\tau$-rigid modules
  14. $F$-invariant of decorated modules
  15. Newton polytopes of $\tau$-rigid modules
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $\mathcal{A}$ be a skew-symmetrizable cluster algebra with initial seed $({\bf x}_{t_0}, B_{t_0})$. Let $u$ and $v$ be two cluster monomials in non-initial cluster variables. If the two $F$-polynomials $F_u^{t_0}$ and $F_{v}^{t_0}$ have the same Newton polytope, then $u=v$.

Theorems & Definitions (104)

  • Definition 1.1: Newton polytopes of polynomials and modules
  • Definition 1.3: $\tau$-rigid module, multi-semibrick and left finite module
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1: Schneider_2013
  • Corollary 2.2
  • proof
  • Definition 2.3: Newton polytopes of polynomials and modules
  • ...and 94 more