The Combinatorics of Affine Algebras and Weyl Groups

这个项目是关于由F代数和Weyl群产生的结构的组合研究。第一个研究对象是仿射李代数的量子化泛包络代数上的模的结晶图。Lusztig和Kashiwara发展了Kac-Moody代数的量子化包络代数上合适模的标准基的深刻而复杂的理论。当量子参数设置为零(低温极限)时，可以得到一个有色有向图，称为模块的结晶图。这个非凡的图几乎编码了该模块的所有重要代数数据。利用结晶图，将许多代数问题归结为组合问题。在+ne Kac-Moody代数的情况下，组合是特别有利的；Kang、Kashiwara、Misra、Miwa、Nakayashima和Nakayashiki证明，晶图的元素可以表示为非常特殊的有限晶图的元素的某些最终周期的有限序列，称为完美晶体。一个目标是明确地确定晶体的有色图结构，这个族的存在是由Hatayama、Kuniba、Okado、Takagi和Y.Y.Yamada猜想的，该族源于对可积系统中Bethe Anatz的研究。另一个目标是给出共形场理论和统计力学中出现的某些重数的显式公式，例如融合系数和分支函数。人们特别感兴趣的是用某种形式(费米子)来表示这样的量，这种形式允许对基本模型的状态进行准粒子解释。这些公式根据A.N.Kirillov和N.Y.Reshetikhin的操纵构型进行了组合描述。第二个研究对象是a+ne Weyl群的Kazhdan-Lusztig(KL)多项式族。对于单李代数，这些多项式在Schubert簇的几何以及Weyl群和单代数群的表示理论中都是突出的；这些现象推广到a+ne代数。一个目标是给出其中某些多项式的显式组合公式(不允许交错和)，这些多项式表现为相关单李代数的不可约模的分次重数，在零锥上的扭函数模中，单李代数的主幂零伴随轨道的闭包。第二个目标是给出A类a+ne Weyl群的某些抛物型KL多项式的公式，它可以用LasCoux，Leclerc和Thibon的带状表来表示。

The project is a combinatorial study of structures arising fromaffine algebras and Weyl groups.The first object of study is the crystal graph of a module over a quantized universal enveloping algebra of an affine Lie algebra. Lusztig and Kashiwara have developed the deep and intricate theory of canonical bases for suitable modules over quantized enveloping algebras of Kac-Moody algebras. When the quantum parameter is set to zero (the \low temperature limit"), one obtains a colored directed graph called the crystal graph of the module. This remarkable graph encodes nearly all the important algebraic data of the module. Using the crystal graph, many algebraic problems are reduced to combinatorial ones. In the case of a+ne Kac-Moody algebras the combinatorics is particularly favorable; it was shown by Kang, Kashiwara, Misra, Miwa, Nakayashima, and Nakayashiki, that the elements of the crystal graph can be expressed as certain eventually periodic finite sequences of elements of very special finite crystal graphs called perfect crystals. In turn, the perfect crystals can be studied using techniques of classical combinatorics such as the theory of Young tableaux.One goal is to determine explicitly the colored graph structure of crystals in a family whose existence was conjectured by Hatayama, Kuniba, Okado, Takagi, and Y. Yamada and which arose from the study of the Bethe Ansatz in integrable systems. Another goal is to give explicit formulae for certain multiplicities that arise in conformal field theory and statistical mechanics, such as fusion coefficients and branching functions. It is of particular interest to express such quantities in a certain form (\fermionic"), one which admits a quasi particle interpretation for the states of the underlying model. Such formulae have combinatorial descriptions in terms of the rigged configurations of A. N. Kirillov and N.-Y. Reshetikhin. The second object of study is the family of Kazhdan-Lusztig (KL) polynomials for a+ne Weyl groups. For simple Lie algebras these polynomials are prominent in the geometry of Schubert varieties and in the representation theory of both the Weyl group and the simple algebraic group; these phenomena generalize for the a+ne algebras. One goal is to give explicit combinatorial (no alternating sums allowed) formulae for certain of these polynomials, which appear as graded multiplicities of irreducible modules for the associated simple Lie algebra, in the modules of twisted functions on the nullcone, the closure of the principal nilpotent adjoint orbit of the simple Lie algebra. A second goal is to give such formulae for certain parabolic KL polynomials for the a+ne Weyl group of type A, which can be expressed in terms of the ribbon tableaux of Lascoux, Leclerc, and Thibon.