Linear Independence Of Cluster Monomials Of The Cluster Algebras Of Type G₂ Via Categorification

Fomin-Zelevinsky hoped that the cluster monomials of cluster algebras could be used as part of a set of canonical basis for quantum groups.So they conjectured that in any cluster algebra,cluster monomials are linearly independent.The conjecture has been proved by Gross-Hacking-Keel-Kontsevich using tropical geometry.This paper mainly uses the method of cat-egorization to verify that the conjecture holds in cluster algebra of G₂type.Specifically,let H be the path algebra corresponding to any quiver with a Dynkin diagram of type E₈as the base map,and mod H is a finitely generated right H-module category,which is an abelian category.LetD^b（mod H）be the bounded derived category of mod H,and its shift functor is denoted asΣ,which also has the Serre functor S,satisfyingΣ14（?）S¹⁵.Consider the self-equivalent functor G:=S⁴Σ^-4,and let_C=D^b（mod H）/G be the orbit category of the derived category D^b（mod H）.The orbit category C is a 2-Calabi-Yau triangulated category,andΣis still used to represent the shift functor inC withΣ⁴（?）id.Starting from an arbitrary cluster tilting object T in_C,the endomorphism algebra of the cluster tilting object is the corresponding cluster tilted algebra of type G₂.When the algebraically closed field k=C,there is a cluster character X_M^T,which gives the one-to-one correspondence from the indecomposable rigid objects in the cat-egory of finitely generated modules of the cluster tilted algebra to the cluster variables of the corresponding cluster algebra of type G₂.This paper uses the above correspondence to prove that Fomin-Zelevinsky’s conjecture holds for the cluster algebra of type G₂.