Comments on Exchange Graphs in Cluster Algebras

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra $\A$ in two cases: when $\A$ is of geometric type and when $\A$ is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of $\A$. Both conjectures were formulated recently by Fomin and Zelevinsky.

We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.

Every two seeds in a field of fractions $\mathcal{F}$ together with a symmetric group element gives rise to an automorphism of $\mathcal{F}$ called an exchange automorphism. For positive cluster algebras, we provide equivalent conditions for exchange automorphisms to be cluster isomorphisms of the corresponding cluster algebras. The equivalent conditions are given in terms of a symmetric group action on the set of seeds, which is also studied. Presentations of groups of clust...

In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of $\mathcal A(\mathcal S)$; (ii) there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster...

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine-learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symm...

For a coefficient free cluster algebra $\mathcal{A}$, we study the cluster automorphism group $Aut(\mathcal{A})$ and the automorphism group $Aut(E_{\mathcal{A}})$ of its exchange graph $E_{\mathcal{A}}$. We show that these two groups are isomorphic with each other, if $\mathcal{A}$ is of finite type excepting types of rank two and type $F_4$, or if $\mathcal{A}$ is of skew-symmetric finite mutation type.

For a fixed seed $(X, Q)$, a \emph{rooted mutation loop} is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called \emph{rooted mutation group} and will be denoted by $\mathcal{M}(Q)$. Let $\mathcal{M}$ be the global mutation group. In this article, we show that two finite type cluster algebras $\mathcal{A}(Q)$ and $\mathcal{A}(Q')$ are isomorphic if and only if their rooted mutation groups are isomorphic and the sets $\mat...

In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchang...

Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give the relations between exchange spectrum of a valued cluster quiver and adjacency spectrum of its underlying valued graph, and between exchange spectra of a valued cluster quiver and its full valued subquivers. The ...

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the non-compact quantum dilogarithm at $b=1$.