Tools for Positivity in Algebraic Combinatorics

代数组合学中的积极性工具

• 批准号: 1600391
• 负责人: Jonah Blasiak
• 金额: $ 19.5万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600391&HistoricalAwards=false
• 关键词: Tools; Positivity; Algebraic; Combinatorics

Nonnegative integer invariants provide an important way of understanding complicated algebraic or geometric objects. Examples include the degree of a polynomial and the number of holes of a surface. The goal of this project is develop general methods to obtain a detailed understanding of nonnegative integer invariants arising in several different areas of mathematics. This may form the foundation for developments in quantum information theory, knot theory, physics, and signal processing. In particular, this project offers potential new insights into the tensor decomposition problem, which is essentially the problem of recovering individual signals from a mixture of signals and has applications in medicine, computer vision, chemistry, and fast matrix multiplication.Positivity problems in algebraic combinatorics ask to find positive combinatorial formulae for nonnegative quantities arising in geometry and representation theory. The goal of this project is to develop tools to solve positivity problems arising in two areas of active research, Macdonald theory and geometric complexity theory. Macdonald polynomials are a two-parameter family of symmetric polynomials, which have ties to many areas including geometry, physics, and knot theory. A major breakthrough in this area came with the proof of the Macdonald positivity conjecture, which showed that important structure coefficients related to Macdonald polynomials are nonnegative. It remains a fundamental open question to give a positive combinatorial interpretation of these coefficients. 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. This project will further develop the theory of noncommutative Schur functions, a powerful tool for solving positivity problems, particularly focusing on applications to Macdonald polynomials and the Kronecker problem.

非负整数不变量为理解复杂的代数或几何对象提供了一种重要的方法。例子包括多项式的次数和表面的孔数。这个项目的目标是开发通用的方法，以获得在几个不同的数学领域产生的非负整数不变量的详细了解。这可能成为量子信息理论、纽结理论、物理学和信号处理的发展基础。 特别是，这个项目提供了潜在的新的见解张量分解问题，这是本质上是从混合信号中恢复单个信号的问题，并在医学，计算机视觉，化学和快速矩阵乘法的应用。代数组合学中的正性问题要求找到几何和表示论中出现的非负量的正组合公式。这个项目的目标是开发工具来解决积极性问题所产生的两个领域的积极研究，麦克唐纳理论和几何复杂性理论。麦克唐纳多项式（Macdonald polynomials）是一个双参数对称多项式族，它与许多领域有联系，包括几何学，物理学和纽结理论。这一领域的一个重大突破是麦克唐纳正性猜想的证明，它表明与麦克唐纳多项式相关的重要结构系数是非负的。它仍然是一个基本的开放问题，给一个积极的组合解释这些系数。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。该项目将进一步发展非交换舒尔函数的理论，这是解决正性问题的有力工具，特别关注麦克唐纳多项式和克罗内克问题的应用。