Algebraic Combinatorics

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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.