quantizing Schur functors

几何复杂性理论是利用代数几何和表示论来研究复杂性理论中的P与NP及相关问题的方法。Kronecker问题是表示理论中的一个基本问题，被认为对这种方法很重要。Kronecker问题要求一个正的组合公式，将对称群的两个不可约表示的张量积分解为不可约。量子群和晶基理论是连接组合学和表示论的有力工具，在过去的几十年里得到了积极的发展。在过去的十年里，Berenstein，Zwicknagl，MulMuley，Sohoni和其他人已经多次尝试使用这个工具来研究Kronecker和相关的多倍体问题。最近，这位研究人员和他的合作者穆尔穆利和索霍尼已经获得了克洛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一方法。更广泛地说，这个项目旨在进一步探索几何复杂性理论开创的代数组合学中复杂性理论和正性之间潜在的深层联系。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。代数组合学是对这样的物体进行计数和组织的研究。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。张量分解问题是一个重要而又困难的问题，涉及到代数组合学、复杂性理论、统计学等诸多领域，在医学、计算机视觉、化学、快速矩阵乘法等领域有着广泛的应用。从本质上讲，它是从混合信号中恢复单个信号的问题。这个项目通过应用代数组合数学中的强大工具，为这个问题提供了潜在的新见解。

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.