Ahmet Seven
In this paper, we give a description of the skew-symmetrizable matrices and their mutation classes which are determined by the generalized Cartan matrices of affine type.
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In this paper, we determine representatives for the mutation classes of skew-symmetrizable 3x3 matrices and associated graphs using a natural minimality condition, generalizing and strengthening results of Beineke-Brustle-Hille and Felikson-Shapiro-Tumarkin. Furthermore, we obtain a new numerical invariant for the mutation operation on skew-symmetrizable matrices of arbitrary size.
A mutation invariant for skew-symmetrizable matrices
本文扩展了Casals针对斜对称矩阵的二进制突变不变量方法，适用于斜对称可解矩阵，并在特定条件下得到两个不同的扩展。
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a c…
In this paper, we study combinatorial properties of quasi-Cartan companions defined by the c-vectors of acyclic skew-symmetrizable cluster algebras. In particular, we show that the diagram of any skew-symmetrizable matrix associated with an acyclic cluster algebra has an admissible cut of edges.
Maximal green sequences are particular sequences of mutations which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Cordova-Vafa in the context of supersymmetric gauge theory. In this paper, we show that skew-symmetrizable 3x3 matrices with a mutation-cyclic diagram do not have any maximal green sequences. We also obtain some properties of maximal green sequences of skew-symmetrizable 3x3 matrices with mutation-acyclic diagrams.
Inspired by Fock-Goncharov's amalgamation procedure \cite{Fock-Goncharov-2006}, Shen-Weng introduced string diagrams in \cite{Shen-Weng-2021}, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. In this paper, we prove that the skew-symmetrizable matrices from string diagrams are in the smallest class $\mathcal P^\prime$ of skew-symmetrizable matrices containing the $1\times 1$ zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, u…
Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in \cite{WG}. As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.
The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices. In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in $\Z$ that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral…
Cluster-cyclic condition of skew-symmetrizable matrices of rank 3 via the Markov constant
本文针对3阶斜对称可化矩阵，结合马尔可夫常数研究其对应加权箭图经突变后始终保持循环的条件，拓展了马尔可夫常数在斜对称矩阵相关研究中的应用场景。
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we aim to generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.
Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$
本文提出了针对特定索引的可对称矩阵的模数Nahm和家族，并构建了相关的向量值自动形式。
In this paper, we obtain relations in the Weyl groups of Kac-Moody algebras that come from mutation classes of skew-symmetrizable matrices. These relations generalize those obtained by Barot and Marsh for finite type. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices.
Mutation Of Matrices Over Group Rings
本文研究了群环上斜对称可化矩阵的突变定义及其与具有对称性的有向无环图的突变关系，并提出了新的突变规则。
Sign-coherence and tropical sign pattern for rank $3$ real cluster-cyclic exchange matrices
本文研究了秩3实簇循环交换矩阵的c向量符号一致性，证明了其与二次方程根的关系，并展示了对应的交换图结构及二面体群的簇实现。
To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph G is the set of all isomorphism classes of graphs that can be obtained from G by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with…
On factorization of matrix of Kazhdan-Lusztig polynomials
本文研究了Hecke代数中Kazhdan-Lusztig多项式矩阵的分解问题，证明其可分解为多个矩阵的乘积，并通过几何方法证明了系数的非负性。
Real $C$-, $G$-structures and sign-coherence of cluster algebras
将簇代数中整数 $C$-, $G$-矩阵理论推广到实情形，通过斜对称化方法将问题约化，并分类了秩2和有限型情况，提出了关于实交换矩阵和 $C$-, $G$-矩阵的两个猜想。
Polytope realization of cluster structures
本文基于多面体函数的变异行为，揭示了多面体面与子簇代数的联系，推广了相容度并构造了包含$g$-扇所有锥的扇$\mathcal{C}$与$\mathcal{N}$，为簇结构提供了新的多面体实现与解释。
A topology on the poset of quiver mutation classes
本文构建了有向图突变类的拓扑结构，揭示了突变不变和继承性质与偏序集下集的对应关系，并探讨了该空间的拓扑性质及未来研究方向。
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