Keys and Demazure crystals for Kac-Moody algebras

Colored five-vertex models and Demazure atoms

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we construct Demazure atoms on Kashiwara's $\mathcal{B}_\infty$ crystal and give new algorithms for computing Lascoux-Schützenberger keys.

半标准杨表左右键的简单计算程序

Simple Procedures for Left and Right Keys of Semi-Standard Young Tableaux

本文提出获取半标准杨表左右键的极简新程序，通过反复执行 Deodhar 极大/极小提升实现，并阐明其与 Demazure 特征标及 LS 路径理论的联系。

广义Galois数通过仿射Kac-Moody代数的渐近性

Asymptotics of generalized Galois numbers via affine Kac-Moody algebras

广义伽罗瓦数计算有限域上向量空间中标志的个数。渐近地，当向量空间的维数变大时，我们给出它们的指数增长性并确定它们的初始值。初始值用θ函数和欧拉母函数表示。我们的渐近计数方法是基于一个Demazure模极限建设的可积最高权表示的仿射Kac-Moody代数。对于计算有限域上向量空间中子空间个数的经典伽罗瓦数，θ函数是雅可比θ函数。我们将我们的研究结果的渐近数的q元代码，并得出结论，通过限制建设仿射卡茨-穆迪代数的渐近枚举有关未来可能的研究与一些最后的意见。

仿射 Demazure 权重多面体与扭曲 Bruhat 序

Affine Demazure Weight Polytopes and Twisted Bruhat Orders

本文定义仿射 Demazure 权重多面体，借助几何不变量理论得到由三类 Bruhat 序指定的定义不等式，并用扭曲 Bruhat 序刻画任意面上的顶点。

Kostant–Kumar模：表示与重数

Kostant--Kumar modules: presentation and multiplicities

该论文研究可对称化Kac–Moody代数上的Kostant–Kumar模，给出其生成元与关系表示，证明其分解数可通过重数空间商的正交投影计算，还得到分解数上界并研究Schur正性。

右钥匙的新刻画，以及t=0处的m对称舒尔函数

A new characterization of right keys, and the $m$-symmetric Schur functions at $t=0$

该研究刻画右钥匙表，发展m对称函数环的舒尔函数在t=0处的组合理论，证明相关柯西恒等式，关联m对称舒尔函数与几乎对称舒尔函数并推导行列式公式。

通过虚拟化实现的关键映射与疏散

Keys and Evacuation via Virtualization

本文证明有限Cartan型晶体的虚拟化与关键映射可交换，将关键映射简化为单李型，还在正交表格中验证相关虚拟化与分裂映射一致，给出正交疏散的纯组合新定义。

Locks fit into keys: a crystal analysis of lock polynomials

Lock polynomials and lock Kohnert tableaux are natural analogues to key polynomials and key Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and show that the difference of a key polynomial and lock polynomial for the same composition is monomial positive. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric. We accomplish these goals combinatorially using key Kohnert tableaux and lock Kohnert tableaux. In particular, for the difference of a key minus a lock, we focus on the behavior of c…

A combinatorial study of affine Schubert varieties in affine Grassmannian

Let $\overline{\mathtt{X}}_λ$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_λ$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $λ$ is a coweight of $G$. We find a simple algorithm which describes the set $Ψ(λ)$ of all $\mathtt{I}$-orbits in $\overline{\mathtt{X}}_λ$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\mathtt{I}$-orbits. These operators satisfy Braid relations generically on the c…

On Combinatorial Models for Affine Crystals

The tableau model for Kirillov-Reshetikhin (KR) crystals, which are finite dimensional crystals corresponding to certain affine Lie algebras, is commonly used for its ease of crystal operator calculations. However, its simplicity makes quite complex the calculation of statistics such as: keys (used to express Demazure characters), the crystal energy function (an affine grading on tensor products of KR crystals), and the combinatorial R-matrix (an affine crystal isomorphism permuting factors in a tensor product of KR crystals). It has been shown that these calculations are much simpler with the…

柏原$b(\infty)$晶体的制备定理

A preparation theorem for the kashiwara $b(\infty)$ crystal

柏原$B(\infty)$晶体属于Kac- Moody李代数的韦尔马模.表面上，它只提供了后者的全局/规范基础的参数化。然而，它更具有丰富的组合结构，从中可以读取任何可积最高权模的相应基的参数化，描述最高权模的张量积的分解，可积最高权模的Demazure子模和Demazure模的平移的Demazure标志。$B(\infty)$一般有无穷多个表示，作为自然数的可数多个副本的子集，每个副本由外尔群元素的连续约化分解给出。在每个表示中，有一个由Kashiwara函数确定的Kashiwara算子的动作。这些函数在条目中是线性的。因此，一个自然的问题是要表明，在每个表示中，子集$B(\infty)$是多面体的。在这里，一个新的方法来解决这个问题的基础上，建立双柏原函数，这是足以表明，后者也是线性的条目。目前的工作解决了两个非常困难的障碍之一，在逐步建设，即所产生的功能必须满足一个总和，或简单的S，条件。它非常微妙地依赖于从前一步获得的函数中出现的系数之间的不等式。唯一剩下的障碍，即足够多的功能，至少可以验证在许多家庭的情况下，虽然这是推迟到随后的文件。这个理论有一些有趣的数字命理学，其中涉及加泰罗尼亚数字在两个不同的方式。

辛Kashiwara-Nakashima表aux的余晶体、辛Willis直接法、虚拟键及其应用

Cocrystals of symplectic Kashiwara-Nakashima tableaux, symplectic Willis like direct way, virtual keys and applications

本文提出了一种计算辛KN表aux左右键的余晶体方法，结合Baker虚拟化嵌入技术，并应用于测试旗簇上辛KN表aux的标准性。