Combinatorial Representation Theory

代数对象通常被认为是复杂和神秘的。在组合表示论中，人们寻求更简单的方法来表示它们，以便理解，组织和应用它们到其他学科。这个项目处理的是表示的张量积，这是一种联合收割机表示的方法。该项目试图理解的主要问题是如何将张量积分解为更简单的表示之和。人们可以把它看作是从混合信号中恢复单个信号的问题。张量积分解是一个难题，但也非常重要，因为它出现在许多领域，包括代数组合学，复杂性理论和统计学，并在医学，计算机视觉，物理，化学，组合表示论中的一个基本的开放问题是在张量积的分解中描述两个不可约表示的重数的对称群;这些多重性被称为克罗内克系数。为这些系数找到一个组合解释的问题已经被标记为组合表示论的主要问题之一。“PI和合作者已经能够将Kronecker系数与分区代数联系起来。这种联系导致了对称群的“通用特征”的发现，这些特征是在单位根处求值时专门针对对称群特征的对称函数。这些对称函数连接了组合表示论中的几个困难问题，并将导致其中几个问题的解决。

Algebraic objects are often perceived as complicated and mysterious. In combinatorial representation theory one seeks simpler ways to represent them in order to understand, organize and apply them to other subjects. This project deals with tensor products of representations, which is a way to combine representations. The main problem that the project seeks to understand is how to decompose tensor products into sums of simpler representations. One can think about it as the problem of recovering individual signals from a mixture of signals. The tensor product decomposition is a difficult problem but also very important, because it shows up in many fields, including algebraic combinatorics, complexity theory, and statistics, and has applications in medicine, computer vision, physics, chemistry, and fast matrix multiplication.A fundamental open problem in combinatorial representation theory is to describe in the decomposition of the tensor product the multiplicities of two irreducible representations of the symmetric group; these multiplicities are called Kronecker coefficients. The problem of finding a combinatorial interpretation for these coefficients has been labeled as one of the "main problems in combinatorial representation theory." The PI and collaborators have been able to connect the Kronecker coefficients to the partition algebra. This connection led to the discovery of the "universal characters" of the symmetric group, which are symmetric functions that specialize to characters of the symmetric group when evaluated at roots of unity. These symmetric functions connect several difficult problems in combinatorial representation theory and will lead to the resolution of several of these problems.