Loop and double loop geometry

环路和双环几何形状

• 批准号: 19K14495
• 负责人: MUTHIAH DINAKAR
• 金额: $ 2.41万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Early-Career Scientists
• 财政年份: 2019
• 资助国家: 日本
• 起止时间: 2019-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-19K14495/
• 关键词: Coulomb branches; KM Affine Grassmannians; KM Affine Hecke Algebras; Iwahori-Hecke algebras; Monopole Operators; Coulomb branch; affine Grassmannian; (double) loop groups; affine Grassmannians; Geometric Satake; p-adic Kac-Moody groups; bow var

One of the long term goals of this project is to better understand the Coulomb branches of quiver gauge theories, which give rise to a definition of Kac-Moody affine Grassmannian slices. Specifically the goal is to understand their geometry explicitly. In November 2022, my collaborator, Alex Weekes, and I posted a preprint titled "Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices". In this paper, we construct embeddings of Kac-Moody affine Grassmannian slices into one another using Fundamental Monopole Operators. In this way, we answer a question posed by Finkelberg in his 2018 address at the ICM (International Congress of Mathematicians). In particular, we are able to construct many Poisson subvarieties of Kac-Moody Affine Grassmannian slices in this way. Our hope is that the Fundamental Monopole Operators will be a crucial tool in further investigations of Kac-Moody Affine Grassmannian slices.Additionally, I have made further progress in the project with Auguste Hebert on understanding completions of Kac-Moody Affine Hecke algebras. I have also made progress in the project with Anna Puskas on the T-basis of Kac-Moody Affine Hecke algebras and its relationship with the length function and Bruhat order.

该项目的长期目标之一是更好地了解Quiver仪表理论的库仑分支，这引起了Kac-Moody Agfine Grassmannian片的定义。特别是目标是明确了解其几何形状。 2022年11月，我的合作者Alex Weekes和我发布了一个标题为“ Kac-Moody Aggine Grassmannian Slices的基本单极操作员和嵌入”的预印本。在本文中，我们使用基本的单极操作员将Kac-Moody Aggine Grassmannian片的嵌入彼此构建在一起。通过这种方式，我们回答了芬克尔伯格在2018年在ICM（国际数学家国际大会）讲话中提出的一个问题。特别是，我们能够以这种方式构建许多Kac-Moody Aggine Grassmannian片的泊松亚体。我们的希望是，基本的单极运营商将成为对Kac-Moody Aggine Grassmannian Slices进行进一步调查的关键工具。在此方面，我在该项目中取得了进一步的进步，奥古斯特·赫伯特（Auguste Hebert）了解了Kac-Moody Affine Hecke代数的完成。我还与安娜·普斯卡斯（Anna Puskas）在Kac-Moody Aggine Hecke代数的T-Basis上取得了进展，及其与长度函数和Bruhat Order的关系。

项目成果

The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman.

定义 A 型仿射格拉斯曼方程以及 Kreiman、Lakshmibai、Magyar 和 Weyman 的猜想。

• 发表时间: 2020
• 期刊: Int. Math. Res. Not. IMRN
• 作者: Muthiah Dinakar;Weekes Alex;Yacobi Oded
• 通讯作者: Yacobi Oded

Toward double affine flag varieties and Grassmannians.

走向双仿射旗变种和格拉斯曼尼亚。

• 发表时间: 2019
• 作者: Dinakar Muthiah;Anna Puskas;Ian Whitehead;Dinakar Muthiah;Dinakar Muthiah;Dinakar Muthiah;Dinakar Muthiah;Dinakar Muthiah;Dinakar Muthiah;Dinakar Muthiah
• 通讯作者: Dinakar Muthiah

Correction factors for Kac-Moody groups and t-deformed root multiplicities

Kac-Moody 群和 t 变形根多重数的校正因子

• DOI: 10.1007/s00209-019-02419-1
• 发表时间: 2019
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Dinakar Muthiah;Anna Puskas;Ian Whitehead
• 通讯作者: Ian Whitehead

Equations for affine Grassmannians and their Schubert varieties

仿射格拉斯曼方程及其舒伯特簇

• 发表时间: 2021
• 作者: Dinakar Muthiah;Anna Puskas;Ian Whitehead;Dinakar Muthiah;Dinakar Muthiah
• 通讯作者: Dinakar Muthiah

University of Queensland/University of Sydney(オーストラリア)

昆士兰大学/悉尼大学（澳大利亚）