Algebraic and combinatorial structures in integrable systems

PI建议研究组合表示理论中的问题，这些问题源于数学物理中的问题，特别是统计力学和保形场理论中的精确可解的二维模型。所研究的表示包括仿射李代数、圈代数和量子仿射代数的有限维模和可积模。讨论的问题包括融合积的Feigin Loktev猜想，Littlewood-Richardson系数的精化(推广)公式，仿射代数的可积模的费米子特征标公式，Feigin-Stoyanovsky结构的精化，以及半无限楔形积。这些组合问题包括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Baxter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不同背景下。最近，在SLE的研究中，分数量子霍尔效应，量子力学中的纠缠模型和弦理论。这些模型催生了量子群的发明，具有显著的组合性质，为研究李代数及其变形的表示性质以及组合和代数恒等式提供了肥沃的土壤。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.