Related papers: Mutation-classes of diagrams via infinite graphs

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of…

In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence…

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups $D_n$, the generalized quaternion groups $Q_n$, the…

We observe that, for each positive integer n > 2, each of the Artin groups of finite type A_n, B_n=C_n, and affine type \tilde A_{n-1} and \tilde C_{n-1} is a central extension of a finite index subgroup of the mapping class group of the…

In this paper, we characterize all the finite dimensional algebras that are derived equivalent to an m-cluster tilted algebra of type A tilde. This generalizes a result of Bobonski and Buan [9].

We give a new simple description of the canonical bijection between the set of triangulations of n-gon and some set of Young diagrams. Using this description, we find flip transformations on this set of Young diagrams which correspond to…

We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In…

Mutation of torsion pairs in triangulated categories and its combinatorial interpretation for the cluster category of Dynkin type $A_n$ and of type $A_\infty$ have been studied by Zhou and Zhu. In this paper we present a combinatorial model…

We present a classification of characteristic Dynkin diagrams for the $A_N$, $B_N$, $C_N$ and $D_N$ algebras. This classification is related to the classification of \cw(\cg,\ck) algebras arising from non-Abelian Toda models, and we argue…

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and…

Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we…

We give an explicit description of the mutation classes of quivers of type \tilde{A}_n. Furthermore, we provide a complete classification of cluster tilted algebras of type \tilde{A}_n up to derived equivalence. We show that the bounded…

Recently a generalization of shifts of finite type to the infinite alphabet case was proposed, in connection with the theory of ultragraph C*-algebras. In this work we characterize the class of continuous shift commuting maps between these…

We use a geometric description of $m$-cluster categories of Dynkin type $A$ to count the the number of coloured quivers in the $m$-mutation class of quivers of Dynkin type $A$. This is related to angulations of polygons and the cell-growth…

We consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we…

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams $E_6, E_7$, $E_8$. These curves are…

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin…