箭图的红数

Combinatorial Auslander-Reiten quivers and reduced expressions

In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\widetilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order $\prec_{[\widetilde{w}]}$ on the subset $Φ(w)$ of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection maps, we can apply their properties to the representation theory of KLR algebras and multiplication structure of dual PBW generators associated to any commutation class $[\widetilde{w}_0]$ of the longest element $w_0$.

Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$

We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts needed are that the existence of a reddening sequence is mutation invariant and passes to induced subquivers. Banff quivers define locally acyclic cluster algebras which are known to coincide with their upper cluster algebras.The existence of reddening sequences for these quivers is consistent with a conjectural relationship between the existence of a reddening sequence and a cluster algebra's equality with its upper…

无限制红规模与符号一致性

Unrestricted Red Size and Sign-Coherence

该研究利用叉证明了连通突变无限箭图的无限制红规模为n-1或n，给出任意箭图无限制红规模的通用公式，并通过初等方法证明两类箭图的c-向量具有符号一致性。

红化序列和无限有向图的突变

Reddening sequences and mutation of infinite quivers

本文提出了一种纯组合框架，用于无限有向图的突变，扩展了红化序列的概念，并统一了拓扑和组合视角。

Building maximal green sequences via component preserving mutations

We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of $T$-systems and $Y$-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by …

Indecomposable invariants of quivers for dimension (2,...,2) and maximal paths, II

An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the reduction to the problem of description of maximal paths satisfying certain condition.