Algebraic and combinatorial structures in integrable systems

PI建议研究组合表示论中的问题，这些问题来自数学物理中的问题，特别是统计力学和共形场论中精确可解的二维模型。表示研究inlcude有限维和可积模的仿射李代数，循环代数，量子仿射代数。解决的问题包括Feigin Loktev猜想的融合产品，公式精制（推广）Littlewood-Richardson系数，费米子字符公式的可积模的仿射代数，改进的Feigin-Stoyanovsky建设，和半无限楔形产品。组合问题包括KR-模的重数系数的某些费米子和公式的恒等式及其推广。该项目的一个组成部分是广义顶点模型和相关函数方程的巴克斯特矩阵的表示理论构造。统计力学和量子场论中的可积模型出现在物理和数学的各种背景下。最近，在SLE的研究中，分数量子霍尔效应，量子力学和弦理论中的纠缠模型。这些模型，这引起了量子群的发明，具有显着的组合性质，并已成为一个肥沃的土壤，研究性质的陈述李代数及其变形，以及组合和代数身份。例如，上面提到的费米子特征公式与分数统计或任意子密切相关。

The PI is proposing to study questions in combinatorial representation theory which arise from problems in mathematical physics, in particular, exactly solvable two-dimensional models in statistical mechanics and conformal field theory. Representations studied inlcude finite-dimensional and integrable modules of affine Lie algebras, loop algebras, and quantum affine algebras. Questions addressed include the Feigin Loktev conjecture for fusion product, formulas for refined (generalizations of) Littlewood-Richardson coefficients, fermionic character formulas for integrable modules of affine algebras, refinements of the Feigin-Stoyanovsky construction, and semi-infinite wedge products. The combinatorial questions include identities for certain fermionic sum formulas for multiplicity coefficients of KR-modules and their generalizations. A component of the project is a representation-theoretical construction of Baxter's matrices for generalized vertex models and the associated functional equations.Integrable models in statistical mechanics and quantum field theory arise in various contexts in physics and mathematics. Most recently, in the study of SLE, the fractional quantum Hall effect, models for entanglement in quantum mechanics and string theory. These models, which gave rise to the invention of quantum groups, have remarkable combinatorial properties and have been a fertile ground for studying properties of representations of Lie algebras and their deformations, as well as combinatorial and algebraic identities. For example the fermionic character formulas mentioned above are intimately related to fractional statistics or anyons.