Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

该项目的重点是研究人员与S.Fomin合作最近发现的簇代数。簇代数是一类特殊的整环，旨在为研究半单群及其表示中的全正性和标准基提供止痛框架。这项研究揭示了与Kac-Moody代数的结构理论、热力学Bethe ansatz、箭图表示和可积系统等不同学科之间的意外联系。这项研究的主要工具之一是多面体组合学，更具体地说，是基于热带演算的分段线性和无减双曲面变换之间的相互作用。这个项目的主要动机来自两个经典的数学领域：表示理论和全正性理论。表象理论是一种研究对称性的数学方法；更具体地说，它编码了自然界中各种物理和生物系统的对称性。全正性是矩阵(数组)的一个显著性质，它推广了常见的正数概念。这两个理论在物理、化学和其他科学以及其他数学学科中都有大量的应用。事实上，表象理论是量子力学的数学基础，而全正性是解释机械系统振荡的主要工具。在过去的十年里，这两个领域之间发现了深刻的联系，它们的应用范围得到了极大的扩展。这个项目探索了表征理论和完全实证性的现代框架，目的是使其形式主义更加明确和易于理解。

The project focuses on cluster algebras recently discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras areintegral domains of a special kind designed to provide analgebraic framework for the study of total positivity andcanonical bases in semisimple groups and their representations.The investigator studies structural properties of cluster algebrasand their quantum deformations. This study uncovers unexpectedconnections with such diverse subjects as the structural theory ofKac-Moody algebras, thermodynamic Bethe ansatz, quiverrepresentations, and integrable systems. One of the maininstruments of the study is polyhedral combinatorics, morespecifically, an interplay between piecewise-linear andsubtraction-free birational transformations based on the tropicalcalculus.The main motivation for this project comes from two classical areasof mathematics: representation theory and the theory of totalpositivity. Representation theory is a mathematical approach tostudying symmetry; more specifically, it encodes the symmetryproperties of various physical and biological systems that occur innature. Total positivity is a remarkable property of matrices (arraysof numbers) that generalizes the familiar notion of positive numbers.Both theoriesfind numerous applications in physics, chemistry and othersciences, as well as in other mathematical disciplines. In fact,representation theory serves as the mathematical foundation ofquantum mechanics, while total positivity is a major tool forexplaining oscillations in mechanical systems. During the lastdecade, deep connections were found between the two fields, andthe scope of their applications was greatly extended. Thisproject explores the modern framework of representationtheory and total positivity, with the goal of making its formalismmuch more explicit and understandable.