Combinatorial aspects of geometry and representation theory

几何与表示论的组合方面

• 批准号: 0301072
• 负责人: Mark Haiman
• 金额: $ 40万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301072&HistoricalAwards=false
• 关键词: Combinatorial; aspects; geometry; representation; theory

Abstract for award of Haiman DMS-0301072Professor Haiman works on problems in combinatorics related toalgebra, geometry and representation theory. Using new discoveries onthe geometry of Hilbert schemes, he has recently proved the positivityconjecture for Macdonald polynomials and established a formula interms of these polynomials for the character of a "doubled" version ofthe classical spaces of harmonics for symmetric groups. Evidencesuggests that these results fit into a larger framework in whichNakajima's quiver varieties play the role of the Hilbert scheme. Inhis continuing research, Haiman hopes to establish the validity ofthis larger framework and further develop the connections betweencombinatorics, Macdonald polynomials and Hilbert schemes. He alsoplans to work on unsolved problems connected with q-analogs ofLittlewood-Richardson coefficients introduced by Shimozono and Weymanand the q-Schur functions of Lascoux, Lapointe and Morse. Finally, incollaboration with Grojnowski, he will return to a study he initiatedsome years ago of Hecke algebra characters and their connection withcombinatorial properties of immanants.Combinatorics is the mathematical study of things that can bedescribed, counted and manipulated in simple and explicit terms:things like trees, Young diagrams, or strings of symbols, that we canwrite on paper or encode in a computer. In my view, what makescombinatorics interesting is that deeply abstract mathematicalconcepts--such as geometric spaces in many dimensions, or algebras ofsymmetries--tend to have an underlying combinatorial structure. Mycurrent work connects the geometric properties of special spaces suchas Hilbert schemes and quiver varieties with the combinatorialproperties of Young diagrams and symmetric polynomials. Byunderstanding the connection, we can deal with these abstract spaceson a concrete level and get important new insights about both thegeometry of the spaces and the rules governing the combinatoricsassociated with them.

摘要海曼奖DMS-0301072海曼教授的作品在组合数学相关的问题togalgebra，几何和表示理论。 他利用Hilbert概型几何的新发现，证明了Macdonald多项式的正猜想，并由此多项式建立了对称群调和函数经典空间的“二重”性质的公式。 证据表明，这些结果适合到一个更大的框架中，Nakajima的类品种发挥作用的希尔伯特计划。 在他的持续研究中，海曼希望建立这个更大框架的有效性，并进一步发展组合学，麦克唐纳多项式和希尔伯特方案之间的联系。 他alsoplans工作未解决的问题与q-类似物的Littlewood-理查森系数介绍了Shimozono和韦曼和q-舒尔功能的Lascoux，Lapointe和莫尔斯。 最后，在与Grojnowski的合作中，他将回到他几年前开始的Hecke代数特征及其与内在组合性质的联系的研究。组合数学是对可以用简单而明确的术语描述、计算和操纵的事物的数学研究：像树、Young图或符号串这样的东西，我们可以写在纸上或在计算机中编码。 在我看来，组合学的有趣之处在于，深度抽象的数学概念--比如多维几何空间或对称代数--往往有一个潜在的组合结构。 我目前的工作是将特殊空间的几何性质，如希尔伯特格式和希尔伯特簇，与Young图和对称多项式的组合性质联系起来。 通过理解这种联系，我们可以在具体的层面上处理这些抽象的空间，并对空间的几何和与之相关的组合学规则有重要的新见解。