Coxeter combinatorics and cluster algebras

S. Fomin和a . Zelevinsky引入聚类代数作为研究半单群中全正基和正则基的框架。此后，它们出现在广泛的数学领域，包括teichmller理论、泊松几何、颤振表示、李论、代数几何、代数组合，甚至在偏微分方程中（在描述浅水波的方程中）。本项目将为聚类代数的研究带来新的柯氏理论工具，以极大地扩展已被人们熟知的聚类代数的范畴，并在有限类型下证明新的结果。大部分研究将使用可分类元素和寒武纪扇的组合学和几何学，这是研究者与D. Speyer合作开发的。这个项目汇集了两个数学研究流，它们都与广泛的数学领域有着深刻的联系。考克斯特群是一种基于反射对称的代数抽象，它们在上个世纪的一些重要数学发展中发挥了作用。簇代数是一个较新的发现，但已经在意想不到的领域显示出惊人的应用。柯赛特理论工具的应用是研究簇代数结构的几种有前途的方法之一。这种相互作用也向另一个方向发展，因为簇代数为理解Coxeter群提供了新的途径。

Cluster algebras were introduced by S. Fomin and A. Zelevinsky as a framework for studying total positivity and canonical bases in semisimple groups. They have since appeared in a wide range of mathematical areas, including Teichmüller theory, Poisson geometry, quiver representations, Lie theory, algebraic geometry, algebraic combinatorics, and even in partial differential equations (in the equations describing shallow water waves). This project will bring new Coxeter-theoretic tools to bear on the study of cluster algebras, in order to greatly expand the class of well-understood cluster algebras, and to prove new results even in finite type. Much of the research will use the combinatorics and geometry of sortable elements and Cambrian fans, developed by the investigator in collaboration with D. Speyer.This project brings together two streams of mathematical research that both have deep connections to a broad array of mathematical fields. Coxeter groups are an algebraic abstraction based on reflective symmetry, and they have played a role in some of the important mathematical developments of the past century. Cluster algebras are a more recent discovery, but have already shown surprising applications in unexpected areas. The application of Coxeter-theoretic tools is one of several promising approaches to the structural study of cluster algebras. The interaction also goes in the other direction, as cluster algebras bring to light new ways of understanding Coxeter groups.