Algebraic Combinatorics

申请人打算继续研究组合方法在代数问题中的应用，以及代数方法在组合枚举中的应用。具体的主题包括研究Coxeter群中的模式回避和约简表达式的枚举，Weyl室环变的上同调中产生的置换字符，使用对称群的群代数研究内不等式，以及研究与部分有序集相关的广义表的生成函数。此外，提议者打算继续开发自由软件来协助这些领域的研究，特别是开发一个用于处理根系统、Coxeter群和半单李代数特征的软件包的新版本。这项研究属于组合学的一般领域。组合学的目标之一是找到研究离散对象集合如何排列的有效方法。离散系统的行为对现代通信极为重要。例如，大型网络的设计，比如那些出现在电话系统中的网络，以及计算机科学中处理离散对象集的算法设计，这就利用了组合研究。

Stembridge 9700787 The proposer intends to continue his research into the applications of combinatorial methods to algebraic problems, and conversely, the use of algebraic methods in combinatorial enumeration. The specific topics include the study of pattern avoidance and enumeration of reduced expressions in Coxeter groups, permutation characters arising in the cohomology of the toric variety of a Weyl chamber, use of the group algebra of the symmetric group to study immanant inequalities, and the study of generating functions for generalized tableaux associated with partially ordered sets. In addition, the proposer intends to continue developing free software to assist research in these areas, and in particular, to develop a new version of a package for working with root systems, Coxeter groups, and characters of semisimple Lie algebras. This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.