A classification of one dimensional affine rank three graphs

仿射不可约Coxeter群的强壳性质

The strong hull property for affine irreducible Coxeter groups of rank 3

本文证明了所有秩为3的仿射不可约Coxeter群满足强壳性质，通过Coxeter复形的几何性质将凸壳分析简化为有限可管理的配置。

小仿射秩3图的自同构群

The automorphism groups of small affine rank 3 graphs

本文研究仿射秩3图的自同构群，通过确定有限多个较小图的完整自同构群并结合已知结果，将一般问题归结为基座为非交换单群的情形。

秩2的下仿射MV多面体

Lower affine MV polytopes of rank 2

本文证明秩2下仿射MV多面体与约化双Bruhat胞腔的非负热带点一一对应，推广了有限型结果，并因反射对称性同样适用于上仿射MV多面体。

如何计算Delaunay多面体的秩

How to compute the rank of a Delaunay polytope

粗略地说，Delaunay多面体的秩（在\cite{DGL92}中首次引入）是其自由度的数量。在\cite{DL}中，利用与$P$有关的超度量，给出了计算Delaunay多面体$P$秩的方法.这里给出了一个更简单更有效的方法，它使用仿射依赖而不是超度量。这种方法适用于经典的Delaunay多面体。然后，我们给出一个没有任何仿射基的Delaunay多面体的例子。

Properties of high rank subvarieties of affine spaces

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry, such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of [5]. We also show that for $k$-varieties $\mathbb X \subset …

仿射晶体、一维和与抛物型Lusztig q-模拟

Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

本文研究了经典仿射型的一维和。我们证明了一个猜想的第三作者和Zabrocki显示他们都分解的一维款项有关的仿射型A提供的秩的根系考虑是足够大的。因此，任何一维和与一个经典的仿射根系具有足够大的秩可以被视为一个抛物Lusztig q-模拟。

Partial linear spaces with a rank 3 affine primitive group of automorphisms

A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on poi…

仿射D型簇范畴非齐次管的几何模型

A geometric model for the non-homogeneous tubes of the cluster category of affine type D

本文针对仿射D型簇范畴的非齐次管，基于双孔圆盘内无向弧的同伦类构建几何模型，并将已有秩n-2管模型扩展至两个秩2管。

非例外型量子仿射代数的稳定Rigged代数

Stable Rigged Configurations for Quantum Affine Algebras of Nonexceptional Types

对于大秩非例外型仿射代数，证明了费米子公式仅依赖于Dynkin图的结点0与其余结点的连接，非A型费米子公式可表示为A型费米子公式与Littlewood-Richardson系数之和.将该结果与math.CO/9901037和arXiv：1002.3715相结合，我们解决了大秩假设下的X=M猜想。

On 2-closures of rank 3 groups

A permutation group $G$ on $Ω$ is called a rank 3 group if it has precisely three orbits in its induced action on $Ω\times Ω$. The largest permutation group on $Ω$ having the same orbits as $G$ on $Ω\times Ω$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.

秩为2的丛代数的突变不变量

Mutation invariants of cluster algebras of rank 2

本文研究秩为2的丛代数的突变不变量，刻画了有限型情形，给出仿射型例子并证明非仿射型Laurent突变不变量不存在，进而将其应用于一类丢番图方程的研究。

Braid relations for involution words in affine Coxeter groups

We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible set of "half-braid" relations to the ordinary braid relations produces a spanning set: in the untwisted case, this occurs for the Coxeter systems which are finite with rank two or type $A_n$, or affine with rank three or type $\tilde A_n$. These results generalize recent work of Hu and Zhang on the finite classical cases.