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

该PI打算进行组合研究的结构所产生的仿射代数与应用representationtheory，数学物理和q series.主要的组合对象是晶体图一方面和操纵配置的另一方面。晶体基为Kashiwara和Lusztig提出的量子化通用包络代数上的模的晶体基的深层理论提供了组合描述：当量子参数q趋于零时，这些基可以通过编码几乎所有基本代数数据的晶体图精确描述。另一方面，扭曲的构型编码了底层物理模型的粒子结构，并导致了费米子公式。本文提出了研究晶体结构和解决长期存在的熔合系数的组合表达式的问题.这些研究将应用于q-级数，特别是Bailey引理，X=M猜想，和临界函数理论.在热力学力学中，有几种方法可以求解某些模型，即通过角转移矩阵法和Bethe Answer.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。对应于角转移矩阵法的指数集中的元素称为晶体基。索引集中对应于Bethe Answer的元素被称为drigged配置。PI提出了研究两个索引集之间的映射，它的性质和推广。这将在数学和物理的许多不同领域中得到应用，如表示论、组合数学和统计力学。

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.