A Method of Classifying Simple Laced Root Systems

Hecke algebras of simply-laced type with independent parameters

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. …

非晶体学关联多面体的多面体实现

Polytopal realizations of non-crystallographic associahedra

本文通过折叠技术证明非单链及非晶体学根系对应的广义关联多面体可由单链广义关联多面体的截面实现。

有限型丛代数的格结构与非单箭袋型Ingalls-Thomas双射

Lattice structure in cluster algebra of finite type and non-simply-laced Ingalls-Thomas bijection

该研究通过显式刻画有限型丛代数的交换箭群，证明其丛集合的格结构与GLS路代数的挠格、$c$-寒武纪格反同构，建立了非单箭袋型的Ingalls-Thomas双射对应。

正交根、Macdonald表示与拟抛物集

Orthogonal roots, Macdonald representations, and quasiparabolic sets

本文研究单laced Weyl群在特定类型下由正交根生成的Macdonald表示，证明极大正交正根集构成拟抛物集，并利用交叉、嵌套和对齐描述其结构，进而给出非嵌套基、非交叉基及偏序集，并用$E_8$的$8$-根简洁描述一个量子同构图。

Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of $\mathcal{F}(w)$ via its connection with the root system. We show that by taking dimension vectors, simple objects in $\mathcal{F}(w)$ bijectively correspond to Bruhat inversion roots of $w$. As an application, we obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the Jordan-Hölder property (JHP). To achieve this, we develop a method to find simple o…