Mathematical Sciences: Combinatorial Aspects of Representation Theory and the Theory of Symmetric Functions

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This award supports the research in algebraic combinatorics of Professors Adriano Garsia and Jeffrey Remmel of the University of California at San Diego. They plan to develop combinatorial algorithms for the construction of representations and their decompositions into irreducible components. Among the specific topics they will work on are: the new Macdonald polynomials and their possible setting in representation theory; the structure of descent algebras of Coxeter groups; and the explicit evaluation of the coefficients in Schur function expansions in certain symmetric formal power series. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. Fundamentally, combinatorics represents a systematization of the very first of all mathematical activities, counting. In its modern development, however, combinatorics has gone beyond just counting to make use of a wide variety of advanced mathematical techniques, and although its roots go back several centuries, the field has had an explosive development in the past few decades because of its importance in communications and information technology.