Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology

研究抽象数学对象的一个基本的、通常是成功的方法是考虑用另一个更具体的对象来表示它的方法。这是一个强有力的想法，代数表示论和相关领域的最新进展为其他领域的转变带来了强大的机会。特别是，几何和组合现象最初特定的表示理论已经出现在许多其他领域，导致有效的新技术和应用。我们的团队处于这些发展的最前沿。PI和五个CoI在过去十年中为重大进展做出了贡献，他们的专业知识范围从代数，几何和拓扑到数学物理。这提供了新的方法来连接代数和几何和拓扑。例子包括分类的Grassmannian集群结构，麦凯对应的反射群，解除李理论的技术，以2维范畴理论，与应用拓扑物理，并推导出分解矩阵的布劳尔代数从广义李几何。在所有情况下，理论之间的插值介质是一个新兴的几何性质，这是没有得到很好的理解。为了促进研究，有一个强烈的需要解释这些现象，并把它们放在一个包容的新范式。因此，我们的建议旨在了解和调查非常不同的领域之间的关系，从而在一个更系统的框架内推进。这一目标将受益于一个广泛的，整体的观点表示理论，包括李群理论，代数几何，低维拓扑和数学物理。我们在利兹的团队是唯一有资格实施这一计划的。与我们的国际合作伙伴机构的许多数学家的专业合作，我们将解决当前的挑战，提供解决方案，以开放的问题，并通过建立桥接到其他领域开发应用程序。我们能够接受纯数学和应用驱动数学的观点，并具有长期服务于物理科学，生命科学和工程需求的潜力。这种观点的统一需要一个程序级的研究结构和代数是这样一个雄心勃勃的企业正确的核心平台。因此，我们的建议将推动最先进的数学，并将塑造我们所触及的领域的未来方向。

A fundamental, and often successful, way of studying an abstract mathematical object is to consider methods of representing it in another, more concrete object. This is a powerful idea, and recent progress in algebraic representation theory and related areas has given rise to strong opportunities for the transformation of other fields. In particular, geometric and combinatorial phenomena initially specific to representation theory have emerged in many other fields, leading to effective new techniques and applications. Our team is at the forefront of these developments. The PI and the five CoIs have contributed to major advances in the past decade, with their expertise ranging from algebra, geometry, and topology to mathematical physics. This provides new ways to link algebra and geometry & topology. Examples include the categorification of the Grassmannian cluster structure, the McKay correspondence for reflection groups, the lifting of Lie-theoretic techniques to 2-dimensional category theory, with applications to topological physics, and the derivation of decomposition matrices of Brauer algebras from generalised Lie geometry. In all cases, the medium for interpolating between the theories is an emergent geometrical property which is not well understood. For the advancement of research, there is a strong need for explaining these phenomena and placing them in an encompassing novel paradigm. Our proposal hence seeks to understand and investigate relations between very different areas, and so to push on from there in a more systematic framework. This aim would benefit from a broad, holistic view of representation theory, embracing Lie theory, algebraic geometry, low dimensional topology and mathematical physics. Our team in Leeds is uniquely qualified to pursue this programme. Together with specialist collaboration of many mathematicians at our international partner institutions, we will address the current challenges, provide solutions to open questions and develop applications by establishing bridging to other fields. We are in a position to embrace the perspectives of both pure and application-driven mathematics, and with the potential, in the long term, for serving the needs of physical sciences, life sciences and engineering. This unification of perspectives requires a programme-level research structure and algebra is the right core platform for such an ambitious venture. Thus our proposal will push forward the mathematical state-of-the-art and will shape the future directions in the areas we touch upon.