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

该奖项支持加州大学圣地亚哥分校的Adriano Garsia教授和Jeffrey Remmel教授在代数组合学方面的研究。他们计划开发组合算法，用于构造表示并将其分解为不可约的组件。他们将研究的具体课题包括：新的麦克唐纳多项式及其在表征理论中的可能设置；Coxeter群的降代数结构；以及某些对称形式幂级数中舒尔函数展开中系数的显式求值。这项研究属于组合学的广泛范畴，这是当今数学中最活跃的领域之一。从根本上说，组合学代表了对所有数学活动——计数——的系统化。然而，在其现代发展中，组合学已经不仅仅是计算，而是利用了各种各样的先进数学技术。尽管它的根源可以追溯到几个世纪以前，但由于它在通信和信息技术中的重要性，该领域在过去的几十年里有了爆炸性的发展。

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.