Some questions in total positivity and cluster algebras

所提出的项目关注于簇代数和全正性。Fomin和Zlevinsky在他们的工作中反复遇到一种新的组合学，他们将其公理化为所谓的簇代数。近年来，簇代数在数学的许多领域都有应用，如箭图表示、非交换几何、离散可积系统、TeichMuller理论和弦理论等。在与Sergey Fomin的联合工作中，PI将探索某些簇代数的张量图模型。公安部还将与林郑月娥共同研究一个密切相关的主题，即完全积极的态度。代数群的全正部分与簇代数和Lusztig正则基密切相关。该研究继续了仿射全正性的研究，这是该领域一个新的且富有成效的方向。表象理论是量子力学和粒子物理研究的重要领域。簇代数为表示论及其相关学科提供了一种新的途径。PI的研究重点是位于我们目前理解的前沿之外的簇代数(所谓的无限突变类型)。与簇代数密切相关的全正性的概念最早出现在上个世纪初的文献中。它是理解机械系统中振动的主要工具。PI的工作集中于将这一理论扩展到仿射(或无限维)情况。

The proposed project focuses on cluster algebras and total positivity. Fomin and Zelevinsky have repeatedly encountered in their work a new kind of combinatorics, which they axiomatized as so-called cluster algebras. Cluster algebras have found applications in numerous areas of mathematics in recent years, such as quiver representations, non-commutative geometry, discrete integrable systems, Teichmuller theory and even string theory. In joint work with Sergey Fomin, the PI will explore a model for certain cluster algebras in terms of tensor diagrams. The PI will also study a closely related topic of total positivity in joint work with Thomas Lam. Totally positive parts of algebraic groups are closely related to cluster algebras and Lusztig's canonical bases. The proposed research continues the study of affine total positivity, which is a new and fruitful direction in this area.The proposed project is rooted in two areas of mathematics: representation theory and combinatorics. Representation theory is an important area that is fundamental for the study of quantum mechanics and particle physics. Cluster algebras provide a new approach to representation theory and related disciplines. The PI's research focuses on cluster algebras that lie just beyond the current frontier of our understanding (the so called infinite mutation type). The notion, closely related to cluster algebras, of total positivity first appeared in the literature at the beginning of previous century. It is a major tool for understanding oscillations in mechanical systems. The PI's work concentrates on extending this theory to the affine (or infinite dimensional) case.