Representation Theory and Double Affine Hecke Algebras

表示论和双仿射赫克代数

• 批准号: 0536962
• 负责人: Bogdan Ion
• 金额: --
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0536962&HistoricalAwards=false
• 关键词: Representation; Theory; Double; Affine; Hecke

he PI intends to undertake a combinatorial study of structuresarising from affine algebras with applications to representationtheory, mathematical physics and q-series.The primary combinatorial objects are crystal graphs on the onehand and rigged configurations on the other hand. Crystal basesprovide a combinatorial description of the deep theory of crystalbases of modules over quantized universal enveloping algebrasdeveloped by Kashiwara and Lusztig: As the quantum parameterq tends to zero, these bases are described precisely by thecrystal graphs encoding nearly all the essential algebraic data.Rigged configurations on the other hand encode the particlestructure of the underlying physical model and lead to fermionicformulas. It is proposed to study the crystal structureon rigged configurations and to tackle the long-standing problemof a combinatorial expression for the fusion coefficients.These studies will have applications to q-series, in particularthe Bailey lemma, the X=M conjecture, and the theory of symmetricfunctions.There are several ways of solving certain models in statisticalmechanics, namely via the corner-transfer-matrix method andthe Bethe Ansatz. Even though the two methods lead to verydifferent looking answers, they describe the same solutions. From the mathematical perspective this suggest that there existsa one-to-one map between the two indexing sets that describe thesolution. The elements in the index set corresponding to thecorner-transfer-matrix method are called crystal bases. The elementsin the index set corresponding to the Bethe Ansatz are calledrigged configurations. The PI proposes to study the mapbetween the two indexing sets, its properties and generalizationsin detail. This will have applications in many diverse areasof mathematics and physics, such as representation theory,combinatorics, and statistical mechanics.

PI 打算对仿射代数产生的结构进行组合研究，并将其应用于表示论、数学物理和 q 级数。主要的组合对象一方面是晶体图，另一方面是操纵配置。晶体基提供了对 Kashiwara 和 Lusztig 开发的量化通用包络代数上模块晶体基的深层理论的组合描述：当量子参数 q 趋于零时，这些基通过编码几乎所有基本代数数据的晶体图来精确描述。另一方面，操纵配置编码了粒子结构 基础物理模型并导出费米子公式。建议研究装配构型上的晶体结构，并解决融合系数的组合表达式这一长期存在的问题。这些研究将应用于 q 级数，特别是贝利引理、X=M 猜想和对称函数理论。解决统计力学中某些模型的方法有多种，即通过角转移矩阵方法和 贝特·安萨兹。尽管这两种方法会得出看起来截然不同的答案，但它们描述了相同的解决方案。从数学角度来看，这表明描述解决方案的两个索引集之间存在一对一的映射。与角转移矩阵方法对应的索引集中的元素称为晶体基。索引集中对应于 Bethe Ansatz 的元素称为操纵配置。 PI 建议详细研究两个索引集之间的映射、其属性和概括。这将在数​​学和物理学的许多不同领域都有应用，例如表示论、组合学和统计力学。