Algebraic Combinatorics and Representation Theory

我的研究项目集中在表示理论、代数组合学和群和半群上的随机游动之间的相互作用。表示的数学概念部分是从对物理对象的对称性的研究中发展出来的。对象的对称性导致关于对象本身的信息。例如，如果两个晶体都表现出相同的本征对称性，那么它们也将表现出相同的物理性质。在这里，我们说这两个晶体都“代表”了对称性。这一观点反映在数学表象理论中。其中一个研究对象表现为给定的对称性集合。这些是对称性的表示。人们通过将它们分解成更小的表示来理解它们。不可分解的表示是构建所有其他表示的构建块。这个分解问题通常是非常困难的。我的大部分研究计划致力于开发分解表示的新技术，并将它们应用于长期存在的分解问题：在概率论中(分解对称随机游动的特征空间)；在群表示理论中(分解高阶Lie模)；在对称函数论中(分解与Stanley-Stembridge e正性猜想相关的上同调表示)。预期的结果包括：提高我们对分解问题的总体理解；通过贡献新技术来提高我们对分解问题的熟练程度；以及在这些长期悬而未决的问题上取得进展。我方法的创新之一利用了代数、组合学和概率论之间的联系。具体地说，我将处理与超平面排列(与一组对称相关的基本组合不变量)上的随机游动相关的代数对象。这一开创性的理论由Bidigare、Hanlon和Rockmore(BHR)首创，并由Brown和Diaconis进一步发展和扩展。上面的分解问题可以在这个理论中重塑，我最近与MarGolis和Steinberg取得的进展允许引入拓扑工具。我的研究计划可以分阶段实施，可以很容易地将硕士和博士生的培养结合起来。它也可以容纳本科生，因为有些部分非常容易处理，但很刺激。此外，我的程序的许多方面都适用于计算机代数的探索。这提供了几个HQP的好处：对研究问题的直觉的有效发展；探索否则很难探索的高水平例子的能力；以及通过高端形式代数操作发展证明策略。

My research program centres on interactions between representation theory, algebraic combinatorics and random walks on groups and semigroups.The mathematical notion of a representation developed, in part, from investigations into the symmetries of physical objects. The symmetries of an object lead to information about the object itself. For example, if two crystals both exhibit the same intrinsic symmetries, then they will also exhibit the same physical properties. Here, we say that both crystals "represent" the symmetries.This idea is reflected in the mathematical theory of representations. One investigates objects that manifest a given collection of symmetries. These are the representations of the symmetries. One understands them by decomposing them into smaller representations. The indecomposable representations are the building blocks from which all others representations can be constructed. This decomposition problem is very difficult in general.A large portion of my research program is dedicated to developing new techniques for decomposing representations, and to applying them to longstanding decomposition problems: in probability theory (decomposing eigenspaces of symmetrized random walks); in group representation theory (decomposing higher Lie modules); in symmetric function theory (decomposing cohomology representations related to the Stanley-Stembridge e-positivity conjecture).The anticipated outcomes include: advancing our general understanding of decomposition problems; improving our proficiency with decomposition problems by contributing new techniques; and making progress on these longstanding open problems.One of the originalities of my approach exploits connections between algebra, combinatorics and probability theory. Specifically, I will work with algebraic objects related to random walks on hyperplane arrangements (a basic combinatorial invariant associated with a set of symmetries). This seminal theory was initiated by Bidigare, Hanlon and Rockmore (BHR) and further developed and extended by Brown and Diaconis. The above decomposition problems can be recast within this theory and my recent advances with Margolis and Steinberg allow the introduction of topological tools. This presents new promising approaches that have already had several successes.My research program can be implemented in phases, and can easily integrate the training of Masters and PhD students. It can also accommodate undergraduate students, since some portions are very tractable yet stimulating. Furthermore, many facets of my program are amenable to computer algebra explorations. This provides several HQP benefits: the effective development of intuition for research problems; the ability to explore high level examples that would be hard to explore otherwise; and the development of proof strategies through high end formal algebra manipulations.