Algebraic Combinatorics

Stembridge 9700787提出者打算继续研究组合方法在代数问题中的应用，反之，代数方法在组合计数中的应用。具体的主题包括研究Coxeter群中约化表达式的模式避免和计数，Weyl腔的Toric簇的上同调中出现的置换特征，利用对称群的群代数来研究内在不等式，以及与偏序集相关的广义表的生成函数的研究。此外，提出者还打算继续开发自由软件以协助这些领域的研究，特别是开发一个新版本的软件包，用于处理根系统、Coxeter群和半单李代数的特征标。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有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

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.