相关论文: Semicanonical basis generators of the cluster alge…

The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for…

We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…

Geiss, Leclerc and Schr\"oer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that…

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a…

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect…

We study Newton polytopes of cluster variables in type A_n cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the…

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…

Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the…

We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of…

We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…

We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the…

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality…

Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in…

Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of…

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 $\widehat{\mathcal{P}}$…

We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine…

Let $Q$ be an affine quiver of type $A_2^{(1)}$. We explicitly construct the cluster multiplication formulas for the quantum cluster algebra of $Q$ with principal coefficients. As applications, we obtain: (1)\ an exact expression for every…

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…

The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type $A_{2}^{(2)}$. As corollaries, we obtain bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases established in [6], and…