Combinatorics and the Dual Canonical Basis

PI将研究量子群，Hecke代数，以及由Kazhdan，Lusztig和Kashiwara引入的特定基的有效描述。这些代数基，便于表示论计算，目前有一些繁琐的递归描述。改进的描述将采用组合解释，非负性质，和簇代数。量子群表示的重要性自20世纪80年代以来得到了广泛的认识，当Sklyanin和其他人发现这些表示为量子场论和统计学中出现的物理问题提供了解决方案时，用不同的数学术语重新表述问题，德林费尔德和金波然后激发了真正巨大的数学和物理学研究量。物理学家贡献了大量的数学成果，并将其应用于研究各种物理现象。 例子包括原子核的旋转和振动，以及最近遗传密码的突变。数学家也利用他们的专业领域不仅帮助解释量子群表示，而且将新知识应用于其他数学问题。PI将继续由考夫曼和其他人发起的工作，通过用线和结的组合图代替某些繁琐的递归程序，有效地描述量子群表示。他还将继续他过去的工作，从这些图中构造数学工具，用于研究对称函数满足的不等式。

The PI will study quantum groups, Hecke algebras, and the efficient description of particular bases of these which were introduced by Kazhdan, Lusztig, and Kashiwara.These algebra bases, which facilitate representation theoretic computations, currently have somewhat cumbersome recursive descriptions.Improved descriptions will employ combinatorial interpretation, nonnegativity properties, and cluster algebras.The importance of quantum group representations has been widely recognized since the 1980s, when Sklyanin and others found that these representations provide solutions to physical problems arising in quantum field theory and statistical mechanics.Reformulating questions in different mathematical terms, Drinfeld and Jimbo then inspired a truly enormous volume of research in mathematics and physics. Physicists contributed substantial mathematical results and applied these to study a variety of physical phenomena. Examples include rotations and vibrations in atomic nuclei, and, rather recently, mutations in genetic code.Mathematicians too used their areas of expertise not only to help explain quantum group representations, but to apply new knowledge to other mathematical problems.The PI will continue work, initiated by Kauffman and others, to efficiently describe quantum group representations by replacing certain cumbersome recursive procedures with combinatorial diagrams of wires and knots.He will also continue his past work to construct, from these diagrams, mathematical tools for studying inequalities satisfied by symmetric functions.