Combinatorial Representation Theory

像群这样的代数对象用来测量对称性的方式与用数字来测量大小的方式相似。在组合表示理论中，我们试图通过将抽象的代数对象与组合对象(如图和矩阵)相关联来使代数对象更容易被访问。组合对象通常更容易理解，更重要的是，它们导致了更高效的计算。在这个提案中，我们感兴趣的是使用组合对象来理解表示的乘积。在这个方案中研究的一个乘积是对称群表示的张量积。我们的目标是设计一种算法，使用组合对象来理解将该乘积分解为更简单的表示。张量积的分解是一个重要的问题，它在代数组合学、复杂性理论和统计学等众多领域都有应用，在医学、计算机视觉、物理、化学和快速矩阵乘法等领域也有应用。从本质上讲，它是从混合信号中恢复单个信号的问题。在组合表示理论中，有三个长期未解决的问题寻求将表示分解为不可约表示。这些问题包括Kronecker问题、Plehysm问题和限制问题。这些问题是相互关联的，在对任何一个问题的理解上取得进展都将导致对其他问题的突破。Zabrocki和PI在Kronecker问题的研究中引入了一种新的对称函数基，这种基源于与分拆代数的联系，并导致了新的组合对象的引入。这种新的对称函数基更好地理解了三个公开问题之间的联系，而引入的组合对象使问题更容易理解。在这份提案中，PI和合作者，包括研究生，将继续使用图代数和对称函数来开发算法，我们希望这些算法将导致对Kronecker问题的理解的进步。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic objects such as groups are used to measure symmetry in a similar way as numbers are used to measure size. In combinatorial representation theory we seek to make algebraic objects more accessible by relating abstract algebraic objects to combinatorial objects such as graphs and matrices. The combinatorial objects are often easier to understand and more importantly they lead to more efficient computation. In this proposal we are interested in using combinatorial objects to understand products of representations. One product investigated in this proposal is the tensor product of representations of symmetry groups. The goal is to devise an algorithm that uses combinatorial objects to understand the decomposition of this product into simpler representations. The decomposition of tensor products is an important problem that has applications to a plethora of fields such as algebraic combinatorics, complexity theory, and statistics, and has applications in medicine, computer vision, physics, chemistry, and fast matrix multiplication. Essentially, it is the problem of recovering individual signals from a mixture of signals. There are three longstanding unsolved problems in combinatorial representation theory that seek to decompose representations into irreducible representations. These include the Kronecker problem, the Plethysm problem and the Restriction problem. These problems are interrelated and making progress in the understanding of any will lead to breakthroughs on the others. Zabrocki and the PI introduced a new basis of symmetric functions that arose from connections to the partition algebra and led to the introduction of new combinatorial objects in the study of the Kronecker problem. This new basis of symmetric functions has provided a better understanding of the connection between the three open problems and the combinatorial objects introduced have made the problems more accessible. In this proposal the PI and collaborators, including graduate students, will continue to develop algorithms using diagram algebras and symmetric functions that we hope will lead to advances in the understanding of the Kronecker problem.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.