The Combinatorics of Affine Algebras and Weyl Groups

该项目是对仿射代数和Weyl群结构的组合研究，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晶体图。Lusztig和Kashiwara发展了Kac-Moody代数的量子化包络代数上合适模的深而复杂的标准基理论。当量子参数被设置为零（低温极限）时，我们得到一个有色的有向图，称为模块的晶体图。这个非凡的图形几乎编码了模块中所有重要的代数数据。利用晶体图，许多代数问题可化为组合问题。在一个+ne的卡茨-穆迪代数的情况下，组合学是特别有利的;它是由康，Kashiwara，Misra，Miwa，Nakayashima和Nakayashiki证明的，晶体图的元素可以表示为某些最终周期有限序列的元素非常特殊的有限晶体图称为完美晶体。反过来，完美的晶体可以用经典组合学的技术来研究，例如Young tableaux理论。一个目标是明确地确定一个家族中晶体的着色图结构，该家族的存在由Hatayama，Kuniba，Okado，Takagi和Y. Yamada和Bethe Answer在可积系统中的研究。另一个目标是给出在共形场论和统计力学中出现的某些多重性的明确公式，例如融合系数和分支函数。用某种形式（“费米子”）来表达这些量是特别有趣的，这种形式允许对基础模型的状态进行准粒子解释。这样的公式具有关于A的操纵构型的组合描述。N.基里洛夫和N.- Y.列谢提欣 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，这些多项式在舒伯特簇的几何学和外尔群和单代数群的表示论中很突出;这些现象推广到a+ne代数。一个目标是给出明确的组合（不允许交替和）公式的某些这些多项式，这似乎是分次重的不可约模块的相关简单李代数，在模块的扭曲功能的零锥，封闭的主要幂零伴随轨道的简单李代数。第二个目标是给这样的公式为某些抛物KL多项式的a+ne Weyl群的类型A，这可以表示在带状tableaux的Lascoux，勒克莱尔，和Thibon。

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.