AFFINE CRYSTALS: COMBINATORICS, ALGEBRA AND GEOMETRY

拟议的研究利用Kashiwara的晶体理论研究仿射Kac-Moody代数。这里的晶体是与可对称化的Kac-Moody代数的每个最高权表示相关联的组合对象(一个集合和一些运算)。柏原的构造大量使用了相关的量子化万能包络代数，但晶体本身通常可以通过其他方法实现。一些这样的实现是纯粹的组合，而另一些则使用非平凡几何。在有限类型中，一种引起人们极大兴趣的实现是使用由Anderson和Kamnitzer开发的Mirkovic-Vilonen(MV)多面体。与他的合作者一起，P.I.目前正在使用箭筒变种为对称仿射类型开发这种组合学的一个版本。仿射MV多面体自从有限类型多面体出现以来就一直在寻找，因此它的构造本身就是一个重要的发展。P.I.将调查这些新的多面体与与仿射代数相关的各种代数和几何结构之间的联系。这项工作还可能导致在所有(不一定是对称的)仿射类型中的MV多面体的概念。这项研究还考虑了从箭图簇的几何中提取其他组合实现的方法，并将晶体理论应用于Macdonald多项式和Demazure特征标。该建议解决了仿射代数理论中的重要问题，并将引起该领域许多人的兴趣。仿射代数在数学物理中很重要，因此有可能产生跨学科的影响。该提案还将为本科生研究项目提供资金，更广泛地说，将有助于培养和发展年轻的数学家。与这项研究相关的几个问题可以从晶体的显式实现的角度进行研究。这些是本科生研究项目的理想选择，因为只需要有限的背景就能解决问题，但它们可以提供一个进入丰富的代表性理论故事的入口点。该提案还将支持学生研讨会，这些研讨会的设计既是为了培训学生的高级学科，也是为了发展他们作为讲演者的技能。最后，P.I.将继续与针对高中生的数学圈等项目合作。

The proposed research studies affine Kac-Moody algebras using Kashiwara's theory of crystals. A crystal here is a combinatorial object (a set along with some operations) associated to each highest weight representation of a symmetrizable Kac-Moody algebra. Kashiwara's construction makes heavy use of the associated quantized universal enveloping algebra, but the crystals themselves can often be realized by other means. Some such realizations are purely combinatorial, and others use non-trivial geometry. In finite type, one realization which has generated a lot of interest uses the Mirkovic-Vilonen (MV) polytopes developed by Anderson and by Kamnitzer. Along with his collaborators, the P.I. is currently developing a version of this combinatorics for symmetric affine types, using quiver varieties. Affine MV polytopes have been sought since the finite type polytopes first appeared, so their construction is itself an important development. The P.I. will investigate the connection between these new polytopes and various algebraic and geometric structures related to affine algebras. This work may also lead to a notion of MV polytope in all (not necessarily symmetric) affine types. The proposed research also considers ways of extracting other combinatorial realizations from the geometry of quiver varieties, and develops applications of crystal theory to Macdonald polynomials and Demazure characters.This proposal addresses important questions in the theory of affine algebras, and will be of interest to a number of people in that field. Affine algebras are important in mathematical physics, so there is potential for cross-disciplinary impact. The proposal will also fund undergraduate research projects, and more generally contribute to the training and developing of young mathematicians. There are several questions related to this research that can be studied in terms of explicit realizations of crystals. These are ideal for undergraduate research projects, as only a limited amount of background is required in order to approach the questions, yet they can provide an entry point into a rich representation-theoretic story. The proposal will also support student seminars which are designed both to train students in advanced subjects and to develop their skills as presenters. Finally, the P.I. will continue to work with programs such as math circles aimed at high school students.