Affine Combinatorics

摘要：本项目的主要前提是仿射组合学的发展，仿射组合学是来自表示论、代数几何和数学物理，特别是仿射晶体理论和仿射舒伯特演算的组合结构。利用Pi和她的合作者新建立的Kirillov-Reshetikhin晶体的组合模型，解决了Hatayama等人的X=M猜想。给出了某些统计力学模型组态和的显式费米子公式。最近还发现，仿射晶体结构在Littlewood-Richardson系数和融合系数的组合表达式以及Macdonald多项式的电荷公式中也是重要的。特别是，Fomin和Greene关于Kirillov-Reshetikhin晶体领域中的非交换对称函数的理论、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还建议研究bi-Hecke么半群和代数的组合表示理论，以及量子R-矩阵，以探索仿射晶体的推广。组合方法通常适用于计算研究。从该项目得出的算法的健壮实现将导致为开源计算机代数系统SAGE开发新的包。PI的研究是在代数组合学领域。这是对离散对象和它们之间的映射(组合学)的研究，以及支配这些对象结构的代数性质。出现的组合问题涉及表象理论(对称性研究)、物理学(粒子的能级及其结构系数)和几何(空间曲线的交点)。这项研究中使用的主要工具之一是结晶图，它描述了这些结构在“零温度极限”下的结构，但又封装了所有重要的性质。由于组合数学的具体性，它非常适合于计算研究。从该项目得出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.