Symmetric functions and vertex operator algebras

该项目将探索EPSRC研究领域的表示论（代数），组合数学和几何以及数学物理之间的联系。研究的中心对象将是对称函数的环，这是所有上述3个领域的重要工具。我们将使用数学物理学中的玻色子-费米子对应，将无限晶格上费米子的构型映射到描述玻色子的对称函数，以探索理论的组合方面，并定义代表量子场论中主要对象的所谓顶点算子。目的和目标：将玻色子-费米子对应推广到新的对称函数类和有限格中，目的是构造新的顶点算子。研究方法的新奇：我们将采用统计力学的模型来描述组合方面。最近的EPSRC国际学科评论强调了加强数学科学学科之间跨学科联系的必要性。我们的项目将做出重要贡献，因为它将涉及来自3个不同领域的技术。可能的合作和影响最有可能是学术性质的，并在论文最后一年计划的研讨会和会议上展示研究成果。

The project will explore connections between the EPSRC research areas of representation theory (algebra), combinatorics and geometry as well as mathematical physics. The central object of study will be the ring of symmetric functions, an important tool in all of the 3 mentioned areas. We will use the boson-fermion correspondence from mathematical physics, which maps configurations of fermions on an infinite lattice to symmetric functions describing bosons, in order to explore combinatorial aspects of the theory and to define so-called vertex operators which represent the main objects in a quantum field theory.Aims and Objectives: Generalise the boson-fermion correspondence to new classes of symmetric functions and for finite lattice with the aim of constructing new vertex operators.Novelty of the research methodology: We will employ models from statistical mechanics to describe the combinatorial aspects.The last EPSRC international subject review has highlighted the need to strengthen cross-disciplinary links between subjects in the mathematical sciences. Our project will make an important contribution as it will involve techniques from 3 different areas. Possible collaborations and impact will be most likely to be of academic nature with presentations of the research at workshops and conferences planned during the final year of the thesis.