AFFINE CRYSTALS: COMBINATORICS, ALGEBRA AND GEOMETRY

本研究利用Kashiwara的晶体理论来研究仿射Kac-Moody代数。晶体在这里是一个组合对象（一个沿着一些操作的集合），与可对称化的Kac-Moody代数的每个最高权重表示相关联。Kashiwara的构造大量使用了相关的量子化通用包络代数，但晶体本身通常可以通过其他方式实现。一些这样的实现是纯粹的组合，和其他使用非平凡的几何。在有限类型中，一个产生了很多兴趣的实现使用了由安德森和卡姆尼策开发的Mirkovic-Vilonen（MV）多面体。沿着他的合作者私家侦探。目前正在开发一个版本的对称仿射类型的组合学，使用numerals品种。自从有限型多面体出现以来，人们一直在寻找仿射MV多面体，因此它们的构造本身就是一个重要的发展。私家侦探将研究这些新的多面体之间的联系和各种代数和几何结构有关的仿射代数。这项工作也可能导致MV多面体的概念，在所有（不一定是对称的）仿射类型。拟议的研究还考虑了从几何学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 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.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。该提案还将资助本科生研究项目，更广泛地说，有助于年轻数学家的培训和发展。有几个问题与这项研究有关，可以在晶体的显式实现方面进行研究。这些都是理想的本科研究项目，因为只有有限的背景是为了解决问题所需的，但他们可以提供一个切入点到一个丰富的代表理论的故事。该提案还将支持学生研讨会，这些研讨会旨在培训学生学习高级科目，并培养他们作为演讲者的技能。最后私家侦探将继续开展针对高中生的数学圈等项目。

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.