quantizing Schur functors

量化 Schur 函子

• 批准号: 1161280
• 负责人: Jonah Blasiak
• 金额: $ 12.85万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161280&HistoricalAwards=false
• 关键词: quantizing; Schur; functors

Geometric complexity theory is an approach to P versus NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for decomposing the tensor product of two irreducible representations of the symmetric group into irreducibles. The theory of quantum groups and crystal bases, which has been actively developed over the last several decades, is a powerful tool for connecting combinatorics and representation theory. In the last decade, there have been several attempts, by Berenstein, Zwicknagl, Mulmuley, Sohoni, and others, to use this tool to study the Kronecker and related plethysm problems. Recently, the investigator and his collaborators Mulmuley and Sohoni have obtained the beginnings of a theory of crystal bases for the Kronecker problem. The main goal of this project is to push this approach further. More broadly, this project aims to further explore the potentially deep connections between complexity theory and positivity in algebraic combinatorics as has been initiated by geometric complexity theory.Objects arising in algebra are typically complicated, mysterious, and have many symmetries. Algebraic combinatorics is the study of counting and organizing such objects. This project will explore some surprising connections between this area and complexity theory (the study of algorithms and their limitations). The tensor decomposition problem is an important and difficult problem that shows up in many fields, including algebraic combinatorics, complexity theory, and statistics, and has applications in medicine, computer vision, chemistry, and fast matrix multiplication. Essentially, it is the problem of recovering individual signals from a mixture of signals. This project offers potential new insights into this problem by applying powerful tools from algebraic combinatorics.

几何复杂性理论（英语：Geometric Complexity Theory）是一种使用代数几何和表示理论来研究复杂性理论中P与NP以及相关问题的方法。 在表示论中的一个基本问题，被认为是重要的这种方法，是克罗内克问题，它要求一个积极的组合公式分解成不可约的对称群的两个不可约表示的张量积。 量子群和晶体基的理论在过去几十年中得到了积极的发展，是连接组合学和表示论的有力工具。 在过去的十年中，Berenstein、Zwicknagl、Mulmuley、Sohoni和其他人曾多次尝试使用这种工具来研究克罗内克和相关的体积描记问题。 最近，研究人员和他的合作者Mulmuley和Sohoni已经获得了克罗内克问题的晶体基础理论的开端。 该项目的主要目标是进一步推动这种方法。 更广泛地说，这个项目旨在进一步探索复杂性理论和代数组合学中的正性之间潜在的深层联系，这是由几何复杂性理论发起的。代数中出现的对象通常是复杂的，神秘的，并且具有许多对称性。 代数组合学是研究计数和组织这些对象的学科。 这个项目将探索这个领域和复杂性理论（算法及其局限性的研究）之间的一些令人惊讶的联系。 张量分解问题是代数组合学、复杂性理论和统计学等许多领域中的一个重要而困难的问题，在医学、计算机视觉、化学和快速矩阵乘法等领域都有应用。 从本质上讲，这是从混合信号中恢复单个信号的问题。 这个项目提供了潜在的新的见解，这个问题，应用强大的工具，从代数组合。