Affine Combinatorics

摘要：本项目的主要前提是发展仿射组合学，它是来自表示理论、代数几何和数学物理的组合结构，特别是仿射晶体理论和仿射舒伯特微积分。利用PI及其合作者新构建的Kirillov-Reshetikhin晶体组合模型，提出解决Hatayama等人的X=M猜想，该猜想给出了某些统计力学模型的组态和的显式费米子公式。最近发现仿射晶体结构在Littlewood-Richardson和聚变系数的组合表达式以及Macdonald多项式的电荷公式中也很重要。特别地，Fomin和Greene在Kirillov-Reshetikhin晶体领域的非交换对称函数理论、仿射nilCoxeter代数和仿射局部plactic代数将被用来寻找这样的公式。此外，PI还提出研究双hecke单群和代数的组合表示理论，以及量子r矩阵来探索仿射晶体的推广。组合方法通常适合于计算研究。从该项目中获得的算法的健壮实现将导致开源计算机代数系统SAGE的新软件包的开发。PI的研究领域是代数组合学。这是一门研究离散对象和它们之间的映射（组合学）以及控制这些对象结构的代数性质的学科。出现的组合问题与表示理论（对称的研究），物理学（粒子的能级及其结构系数）和几何（空间曲线的相交）有关。本研究中使用的主要工具之一是晶体图，它在“零温度极限”下描述这些结构，但却捕获了所有重要的性质。由于它的具体性，组合学非常适合于计算研究。从这个项目中得到的算法将在开源计算机代数系统SAGE中实现。

Abstract:The main premise of this project is the development of affine combinatorics, which are combinatorial structures coming from representation theory, algebraic geometry, and mathematical physics, in particular affine crystal theory and affine Schubert calculus. Using the newly constructed combinatorial models for Kirillov-Reshetikhin crystals by the PI and her collaborators, it is proposed to tackle the X=M conjecture of Hatayama et al. which gives explicit fermionic formulas for configuration sums of certain statistical mechanical models. Affine crystal structures have recently also been found to be important in combinatorial expressions for Littlewood-Richardson and fusion coefficients, as well as charge formulas for Macdonald polynomials. In particular, Fomin and Greene's theory of noncommutative symmetric functions in the realm of Kirillov-Reshetikhin crystals, the affine nilCoxeter algebra, and the affine local plactic algebra will be used to find such formulas. In addition, the PI proposes to study the combinatorial representation theory of bi-Hecke monoids and algebras, and the quantum R-matrix to explore generalizations of affine crystals. Combinatorial methods are often amenable to computational investigations. The robust implementation of algorithms derived from the project will lead to the development of new packages for the open-source computer algebra system SAGE.The PI's research is in the area of algebraic combinatorics. This is the study of discrete objects and maps between them (combinatorics) together with algebraic properties that govern the structures of these objects. The combinatorial problems which arise are related to representation theory (study of symmetries), physics (energy levels of particles and their structure coefficients), and geometry (intersections of curves in space). One of the main tools used in this research are crystal graphs, which describe these structures in the "zero temperature limit" and yet encapture all of the important properties. Due to its concreteness, combinatorics is very amenable to computational investigations. The algorithms derived from this project will be implemented in the open-source computer algebra system SAGE.