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Log-concavity and unimodality of cluster monomials of type $A_3$

Zhichao Chen

Abstract

The log-concavity of cluster variables of type $A_n$ and cluster monomials of type $A_2$ was established by Chen-Huang-Sun. It is still a conjecture for the cluster monomials of higher rank. In this paper, we prove the log-concavity and unimodality of the cluster monomials of type $A_3$, a substantially more intricate case. Moreover, we refine and extend this conjecture by considering the unimodality and the strongly isomorphism of cluster algebras.

Log-concavity and unimodality of cluster monomials of type $A_3$

Abstract

The log-concavity of cluster variables of type and cluster monomials of type was established by Chen-Huang-Sun. It is still a conjecture for the cluster monomials of higher rank. In this paper, we prove the log-concavity and unimodality of the cluster monomials of type , a substantially more intricate case. Moreover, we refine and extend this conjecture by considering the unimodality and the strongly isomorphism of cluster algebras.
Paper Structure (15 sections, 18 theorems, 47 equations, 4 figures, 3 tables)

This paper contains 15 sections, 18 theorems, 47 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Seeds and cluster algebras
  4. Permutation and compatibility
  5. A combinatorial framework
  6. Log-concave and unimodal Laurent polynomials
  7. Binomial coefficients and the convolution
  8. Classification of the cluster monomials of type $A_3$
  9. Log-concavity and unimodality of cluster monomials of type $A_3$
  10. Main theorem
  11. Proof of the main theorem
  12. Inward type
  13. Straightforward type
  14. Cyclic type
  15. A refined conjecture

Key Result

Proposition 1.2

The cluster variables of type $A_n$ and the cluster monomials of type $A_2$ are unimodal.

Figures (4)

  • Figure 1: Dynkin diagram of type $A_3$
  • Figure 2: Non-isomorphic quivers of type $A_3$
  • Figure 3: Three reduced cases of type $A_3$
  • Figure 4: The generalized associahedra of type $A_3$

Theorems & Definitions (49)

  • Conjecture 1.1: \ref{['An conj']}
  • Proposition 1.2: \ref{['prop: A3 unimodal']}
  • Proposition 1.3: \ref{['prop: classification']}
  • Theorem 1.4: \ref{['thm: main result']}
  • Conjecture 1.5: \ref{['general conj']}
  • Definition 2.1: Cluster algebra
  • Theorem 2.2: FZ02, GHKK18
  • Definition 2.3: Finite type
  • Definition 2.4: Cluster monomial
  • Remark 2.5
  • ...and 39 more