Combinatorial Aspects of Representation Theory, Mathematical Physics and q-Series

PI打算对仿射代数的结构进行组合研究，并将其应用于表示理论、数学物理和q系列。主要的组合对象一方面是晶体图，另一方面是操纵构型。晶体基提供了由Kashiwara和Lusztig开发的量子化通用包络代数上模的晶体基的深层理论的组合描述：当量子参数趋于零时，这些基被编码几乎所有基本代数数据的晶体图精确地描述。另一方面，操纵的配置编码了底层物理模型的粒子结构，并导致费米子公式。提出了在操纵构型下研究晶体结构的方法，并解决了长期存在的融合系数组合表达式的问题。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。在统计力学中，求解某些模型有几种方法，即角传递矩阵法和Bethe Ansatz法。尽管这两种方法得到的答案看起来非常不同，但它们描述的是相同的解决方案。从数学的角度来看，这表明在描述解决方案的两个索引集之间存在一对一的映射。与角转移矩阵法相对应的索引集中的元素称为晶体基。索引集中与Bethe Ansatz相对应的元素称为赋值配置。PI建议详细研究两个索引集之间的映射，其性质和推广。这将在数学和物理的许多不同领域得到应用，例如表示理论、组合学和统计力学。

The 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.