Discrete Quantum Integrability, Quantum Q-Systems, and Generalized Macdonald Operators

这个研究项目研究数学和理论物理的交叉问题。该项目的重点是源于统计力学的问题，统计力学描述了具有大量组件的系统的平均行为。对这些系统及其对称性的代数研究导致了新的代数，其结构反过来由作为方程组解的函数集合组合编码。这些新的代数，它们的推广，以及由此产生的组合是研究的主要主题。这项研究将与研究生和本科生一起进行，并将加强我们对这些研究领域如何相互联系的理解。更准确地说，该项目涉及循环电流代数模块的梯度张量积，特别是Kirillov-Reshetikhin模块。组合框架是相关q系统的量子聚类代数，其解可以用平面平铺和非相交路径模型表示。通过研究这些系统的离散可积性，研究者期望产生守恒量，迄今为止只在a型中实现。反过来，这些应该导致张量积积的梯度特征的差分方程，它推广了差分Toda方程。此外，这些方程的解可以表示为产生差算子的迭代作用，这些差算子与在双仿射Hecke代数的多项式表示范围内定义的广义Macdonald算子密切相关。与量子仿射代数和椭圆霍尔代数的关系也将被研究，以及更多的组合问题，如非相交路径枚举的Gessel-Viennot行列式的一些（量子）非交换对应物的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project investigates questions at the intersection of mathematics and theoretical physics. The project is focused on questions that originate in statistical mechanics, which describes the average behavior of systems with large numbers of components. The algebraic study of these systems, and their symmetries, leads to new algebras, whose structure is in turn encoded combinatorially by collections of functions that arise as solutions to systems of equations. These new algebras, their generalizations, and the resulting combinatorics are the main topics to be studied. The research will be conducted in association with students at the graduate and undergraduate level, and will enhance our understanding of how these fields of study connect to each other.More precisely, the project concerns graded tensor products of cyclic current algebra modules, specifically Kirillov-Reshetikhin modules. The combinatorial framework is the quantum cluster algebra of the associated Q-systems, the solutions of which can be expressed in terms of plane tilings and non-intersecting path models. By investigating the discrete integrability of these systems, the investigators expect to produce conserved quantities, so far achieved only in type A. In turn, these should lead to difference equations for the graded characters of tensor product product, which generalize the difference Toda equations. Moreover, the solutions of these equations can be expressed as iterated action of difference creation operators, closely related to generalized Macdonald operators defined within the context of polynomial representations of double affine Hecke algebras. Relations to quantum affine and elliptic Hall algebras will also be investigated, as well as more combinatorial questions such as the existence of some (quantum) non-commutative counterpart to the Gessel-Viennot determinant for non-intersecting path enumeration.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.