Algebraic Combinatorics

Stembridge和Fomin的DMS- 0245385提案摘要研究人员将继续研究代数、表示论和代数几何中出现的组合结构，并开发软件工具来帮助这些领域的研究人员。这样的结构的一个例子是簇代数，旨在提供一个具体的代数框架对偶canonicalbases和全积极的半单群。我们的研究还将包括广义Kostka-Foulkes多项式，子Coxeter安排，以及晶体图和广义关联面体的组合学的研究。在选择组合问题和结构进行研究时，提议者的意图是确定最有可能提供其他学科洞察力的领域。事实上，组合数学在数学和计算机科学中的应用越来越广泛，它扮演着重要的角色;这种日益增长的重要性与对离散结构的明确或算法理解的需求有关，并且随着计算在数学研究的几乎所有阶段中的突出地位而增长。

Abstract for the proposal DMS- 0245385 of Stembridge and FominThe investigators will continue their study of combinatorialstructures arising in algebra, representation theory and algebraicgeometry, and to develop software tools to assist researchers inthese areas. One example of such structures are the cluster algebras,designed to provide a concrete algebraic framework for dual canonicalbases and total positivity in semisimple groups. Our investigationswill also include the study of generalized Kostka-Foulkes polynomials,sub-Coxeter arrangements, and the combinatorics of crystal graphs andgeneralized associahedra.In selecting combinatorial problems and conjectures to investigate,the proposers intent has been to identify areas that are most likely toprovide insight into other disciplines. Indeed, combinatorics has animportant role to play in an ever widening array of applications inboth mathematics and computer science; this increasing importance isrelated to the demand for explicit or algorithmic understanding ofdiscrete structures, and has grown in tandem with the prominence ofcomputation in nearly all phases of research in mathematics.