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.
