Representations of p-adic Covering Groups and Integrable Systems

p-adic 覆盖群和可积系统的表示

基本信息

批准号：
2101392
负责人：
Benjamin Brubaker
金额：
$ 29.5万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101392&HistoricalAwards=false
关键词：
Representations adic Covering Groups Integrable

项目摘要

This project will establish new connections between mathematics and physics. On the mathematics side, it explores objects originating from number theory that possess a sophisticated collection of symmetries. On the physics side, it involves lattice models from statistical mechanics. These lattice models are attempts to study the behavior of matter (e.g., its phase transitions from solid to liquid to gas) by determining the energy in each individual atomic interaction. While such a reductionist physical approach may seem daunting, it works surprisingly often and, roughly speaking, the models which succeed in this approach are termed "solvable." In this project solvable lattice models are used to represent special functions from number theory and to demonstrate previously unknown properties of them and the connections between mathematics and physics serve to inform both subjects. The project will provide training opportunities for undergraduate and graduate students that are broadly applicable to a wide range of STEM careers. More precisely, the goal of the project is to provide algebraic structure to the study of matrix coefficients on groups over local fields and their covers. The PI and his collaborators have demonstrated a fundamental connection between metaplectic Whittaker functions on p-adic groups and quantum groups, using the solvable lattice models described above as a bridge between the two theories. The research will demonstrate that this connection extends and generalizes in multiple directions, and a solvable lattice model for Iwahori fixed vectors in a metaplectic Whittaker model is now within reach. Once completed, the project will provide a powerful connection between metaplectic representations and quantum affine superalgebras via the R-matrices used to demonstrate Yang-Baxter equations. Additional topics to be investigated include finding direct links between p-adic representation theory and quantum groups; using Hecke algebra characters to categorize matrix coefficients to better understand unramified calculations in the local theory, which are essential in integral representations of L-functions; and describing enhancements to the theory of Kashiwara crystals for superalgebras and to archimedean matrix coefficients on geometric crystals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将在数学和物理之间建立新的联系。在数学方面，它探索源自数论的具有复杂对称性集合的对象。在物理方面，它涉及统计力学的晶格模型。这些晶格模型试图通过确定每个原子相互作用的能量来研究物质的行为（例如，从固体到液体到气体的相变）。虽然这种还原论的物理方法可能看起来令人畏惧，但它的效果却令人惊讶，粗略地说，在这种方法中成功的模型被称为“可解”。在该项目中，可解晶格模型用于表示数论中的特殊函数，并证明它们以前未知的属性，以及数学和物理之间的联系为这两个学科提供信息。该项目将为本科生和研究生提供广泛适用于各种 STEM 职业的培训机会。更准确地说，该项目的目标是为研究局部域及其覆盖范围上的群的矩阵系数提供代数结构。 PI 和他的合作者使用上述可解晶格模型作为两种理论之间的桥梁，证明了 p 进群和量子群上的元波惠特克函数之间的基本联系。该研究将证明这种联系在多个方向上延伸和推广，并且在metaplectic Whittaker模型中Iwahori固定向量的可解晶格模型现在已经触手可及。一旦完成，该项目将通过用于演示 Yang-Baxter 方程的 R 矩阵，在元波表示和量子仿射超代数之间提供强大的连接。其他要研究的主题包括寻找 p-adic 表示理论和量子群之间的直接联系；使用赫克代数特征对矩阵系数进行分类，以更好地理解局部理论中的无分支计算，这对于 L 函数的积分表示至关重要；并描述了超代数柏原晶体理论和几何晶体阿基米德矩阵系数的增强。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。