arXiv subjects
Publications and source records attributed to Peigen Cao.
We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra $\mathcal A(\mathcal S)$ is just an automorphism of the ambient field $\mathcal F$ which restricts to a permutation of cluster variables of $\mathcal A(S)$.
In this paper, we introduce the enough $g$-pairs property for a principal coefficients cluster algebra, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough $g$-pairs property. As an application, we prove the positivity of denominator vectors for any skew-symmetrizable cluster algebra. In fact, we give complete answers to some long standing conjectures on denominator vectors of cluster variables (see Conjecture 1.1 below), which are proposed by Fomin and Zelevinsky in [Compos. Math. 143(2007), 112-164]. In addition, we prove that the seeds whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of this cluster algebra. Lastly, a criterion to distinguish whether particular cluster variables belong to one common cluster is given.
In this paper, we prove that positivity of denominator vectors holds for any skew-symmetric cluster algebra.
In this paper, we prove that each matrix in $M_{m\times n}(\mathbb Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$ skew-symmetrizable integer matrix. Using such matrices, we introduce the definition of irreducible skew-symmatrizable matrix. Based on this, the existence of a maximal green sequence for a skew-symmetrizable matrices is reduced to the existence of a maximal green sequence for irreducible skew-symmetrizable matrices.
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras) from the perspective of Laurent phenomenon. In this general framework, we firstly prove that each positive and d-vector-positive GLP algebra has the proper Laurent monomial property and thus its cluster monomials are linearly independent. Skew-symmetric cluster algebras are verified to be d-vector-positive, which gives the affirmation of Conjecture [conjd] in [FZ3] in this case. And since the positivity of skew-symmetric cluster algebras is well-known, the new proof is obtained for the linearly independence of cluster monomials of skew-symmetric cluster algebras. For a class of GLP algebras which are pointed, g-vectors of cluster monomials are defined. We verify that for any positive GLP algebra pointed at $t_0$, the g-vectors ${\bf g}_{1;t}^{t_0},\cdots, {\bf g}_{n;t}^{t_0}$ form a $\mathbb Z$-basis of $\mathbb Z^n$, and different cluster monomials have different g-vectors. As a direct application, we verify that Conjecture [conjg] in [FZ3] holds for skew-symmetrizable cluster algebras and acyclic sign-skew-symmetric cluster algebras.
For a skew-symmetrizable cluster algebra $\mathcal A_{t_0}$ with principal coefficients at $t_0$, we prove that each seed $Σ_t$ of $\mathcal A_{t_0}$ is uniquely determined by its {\bf C-matrix}, which was proposed by Fomin and Zelevinsky in \cite{FZ3} as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign-coherence of $c$-vectors hold for $\mathcal A_{t_0}$, which was actually verified in \cite{GHKK}. More discussion is given in the sign-skew-symmetric case so as to obtain a conclusion as weak version of the conjecture in this general case.
Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the ${\bf g}$-vectors of any cluster form a basis of $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(Σ)$ is given, which is realized as $(\mathcal C^{\widetilde Q},Γ)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructed from an unfolding $(Q,Γ)$ of the acyclic exchange matrix $B$ of $\mathcal A(Σ)$.
In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.