Combinatorial Representation Theory

本文提出了一组问题，旨在整合组合表示理论中三个看似不同的领域：中心化代数、Kronecker积和对称群代数的子代数。组合表示理论中的一个基本开放问题是描述对称群的两个不可约表示的张量积分解的多重性。这些多重系数被称为克罗内克系数。寻找这些系数的组合解释的问题已经开放了近一百年，是组合表示理论的主要问题之一。在一个新的发展中，PI和合作者建立了这个问题与对称群的一个中心化代数——划分代数的新联系。PI将利用这种新的联系来研究与克罗内克系数相关的几个突出的开放问题。PI还将继续她在组合代数方面的贡献，组合代数是可以通过组合方法描述的代数，例如下降代数和峰值代数。下降代数的意义在于它们在非交换和拟对称函数中的应用，以及它们与对称群表示环的联系。在这个项目中，PI建议研究新的组合代数，并使用中心化代数来更好地理解现有的组合代数。这项研究属于代数组合学的一般范畴。代数对象通常是复杂而难以理解的。组合学有助于使抽象的代数对象更加具体，从而使它们变得更容易接近，并且可以更容易地用于其他领域。这是通过有效的算法和基本模型来实现的，这些模型可以洞察代数对象的属性。拟议的项目将解决深刻而重要的问题，这些问题将促进对许多领域的理解，包括量子信息论、复杂性理论和算法。研究结果将应用于从物理、化学到计算机科学的许多领域。此外，所提出的研究对研究对称函数、李论、量子群、代数几何、低维拓扑和数学物理的人具有广泛的兴趣。

This proposal brings together a set of problems aimed at integrating three seemingly different areas in combinatorial representation theory: centralizer algebras, Kronecker products and subalgebras of the symmetric group algebra. A fundamental open problem in combinatorial representation theory is to describe the multiplicities in the decomposition of the tensor product of two irreducible representations of the symmetric group. These multiplicities are called the Kronecker coefficients. The problem of finding a combinatorial interpretation for these coefficients has been open for nearly a hundred years and is one of the main problems in combinatorial representation theory. In a new development the PI and collaborators have established a new connection of this problem to the partition algebra, a centralizer algebra of the symmetric group. The PI will use this new connection to study several outstanding open problems related to the Kronecker coefficients. The PI will also continue her contributions to combinatorial algebras, which are algebras that can be described via combinatorial methods, e.g., the descent and peak algebras. The significance of the descent algebras arise from their applications to noncommutative and quasi-symmetric functions, and from their connections to the ring of representations of the symmetric group. In this project, the PI proposes to investigate new combinatorial algebras and to use centralizer algebras to better understand the existing combinatorial algebras.This research is in the general area of algebraic combinatorics. Algebraic objects are often complex and difficult to understand. Combinatorics help make abstract algebraic objects more concrete so that they become more accessible and can be more easily used in other fields. This is achieved through efficient algorithms and elementary models which give insight into the properties of algebraic objects. The proposed projects tackle deep and important problems that will advance the understanding of many fields, including quantum information theory, complexity theory, and algorithms. The results will have applications that would span many fields from physics and chemistry to computer science. In addition, the proposed research is of broad interest in mathematics to those studying symmetric functions, Lie theory, quantum groups, algebraic geometry, low dimensional topology and mathematical physics.