W-Symmetry of the Adelic Grassmannian

有限维代数的阿德尔与有理格拉斯曼流形

Adelic and Rational Grassmannians for finite dimensional algebras

本文针对任意有限维复代数 R 建立阿德尔与有理格拉斯曼流形理论，用不可分解投射模和本原幂等元给出等价刻画，证明其分类 R 值指数函数的双谱 Darboux 变换，并基于完美配对构造两类格拉斯曼流形间的典范嵌入。

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture

We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [M…

Cluster duality and mirror symmetry for Grassmannians

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $\mathbb X=Gr_{n-k}(\mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(\check{\mathbb X}^\circ, W_q:\check{\mathbb X}^\circ \to \mathbb{C})$, where $\check{\mathbb X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{\mathbb X} = Gr_k((\mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expr…

Affine Type $A$ Geometric Crystal on the Grassmannian

We construct a type $A_{n-1}^{(1)}$ geometric crystal on the variety ${\rm Gr}(k,n) \times \mathbb{C}^\times$, and show that it tropicalizes to the disjoint union of the Kirillov-Reshetikhin crystals corresponding to rectangular tableaux with $n-k$ rows. A key ingredient in our construction is the $\mathbb{Z}/n\mathbb{Z}$ symmetry on the Grassmannian coming from cyclically shifting the basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to combinatorial promotion. Additionally, we use the loop group ${\rm GL}_n(\mathbb{C}(λ))$ to define a unipoten…

格拉斯曼流形的范畴化与镜像对称

Categorification and mirror symmetry for Grassmannians

本文研究格拉斯曼流形的范畴化与镜像对称，定义两类簇特征多项式，建立其与突变的关系，通过$g$-向量幺半群实现镜像对称的范畴化，相关机制可推广至正构子簇变体。

聚类深位点与镜像对称

Cluster deep loci and mirror symmetry

本文提出聚类簇深位点的稳定子刻画猜想，用Demazure编织在有限聚类型及Grassmann正电子层中验证，并推出Fukaya范畴有限分裂生成与同调镜像对称。