Standard bases for affine SL(n)-modules
Standard bases for affine SL(n)-modules
复制标题
仿射 SL(n) 模块的标准底座
DOI：
10.1155/imrn.2005.1251
复制
发表时间：
2004
影响因子：
1
通讯作者：
J. Weyman
中科院分区：
文献类型：
--
作者：
V. Kreiman;V. Lakshmibai;P. Magyar;J. Weyman
本文给出了仿射李代数基本表示中Demazure模的一个初等且易于计算的基！sln（和循环组“SLn”）。一个新的特点是，我们定义我们的基础“自下而上”提高每个极值权重向量，而不是“自上而下”降低最高的权重向量。我们的基础自然产生的组合的索引集，其中包括某些子集的整数第一指定的Jimbo等。在晶体操作者方面。我们给出了一个新的方法来定义这些特殊的集合的递归，但非常简单的算法，屋顶运营商，这是类似于左关键建设的Lascoux Schutzenberger。屋顶算子在某种意义上与水晶算子正交。仿射Kac-Moody代数lize ln（或圈群<$Ln）最重要的表示是基本表示V（Λ0），它是与扩展Dynkin图An −1的额外节点相关联的最高权表示。无限维空间V（Λ0）被有限维Demazure模Vw（Λ0）过滤，其中w是仿射Weyl群的元素：它们是loop群的Borel子群的模。不可约表示及其Demazure模有几种一般的构造，例如Lusztig的标准基和Littelmann的收缩模。然而，它们是非常难以计算明确，甚至组合索引集的基础是非常复杂的。我们将给出V（Λ0）及其Demazure模的一个初等且易于计算的基。我们在Fock空间F中工作，F是一个包含V（Λ0）的无限楔形积，类似于实现SLnC基本表示的空间Σ C。Fock空间有一个自然基，由整数的某些无限子集索引。我们的问题的组合部分相当于定义这些子集中的哪些子集将对给定w的Vw（Λ0）的基元素进行索引。我们描述这些特殊的子集的递归，但非常简单的算法，屋顶运营商的子集。这类似于Lascoux-Schutzenberger [11]的左键构造，其区分了对SLnC的给定Demazure模的基的Young tableaux索引。屋顶算子比晶体图算子更基本（也更有效），并且在某种意义上与它们正交。我们可以将屋顶算子看作是跨越晶体图的跳跃，将每个顶点向下移动到极值权重顶点w（Λ0），但不是沿着晶体图的沿着边。与Raghavan-Sankaran [16]的方法类似，屋顶算子的组合学自然地导致我们标准基的定义。一个新的特点是，我们定义我们的基础“自下而上”，提高每个极值权重
We give an elementary and easily computable basis for the Demazure modules in the basic representation of the affine Lie algebra ! sln (and the loop group " SLn). A novel feature is that we define our basis “bottom-up” by raising each extremal weight vector, rather than “top-down” by lowering the highest weight vector. Our basis arises naturally from the combinatorics of its indexing set, which consists of certain subsets of the integers first specified by Jimbo et. al. in terms of crystal operators. We give a new way of defining these special sets in terms of a recursive but very simple algorithm, the roof operator, which is analogous to the left-key construction of Lascoux-Schutzenberger. The roof operator is in a sense orthogonal to the crystal operators. The most important representation of the affine Kac-Moody algebra ŝln (or of the loop group ŜLn) is the basic representation V (Λ0), the highest-weight representation associated to the extra node of the extended Dynkin diagram A n−1. The infinite-dimensional space V (Λ0) is filtered by the finite-dimensional Demazure modules Vw(Λ0) for w an element of the affine Weyl group: these are modules for a Borel subgroup of the loop group. There are several general constructions for irreducible representations and their Demazure modules, such as Lusztig’s canonical basis and Littelmann’s contracting modules. However, they are extremely difficult to compute explicitly, and even the combinatorial indexing set for a basis is very intricate. We will give an elementary and easily computable basis for V (Λ0) and its Demazure modules. We work inside the Fock space F , an infinite wedge product which contains V (Λ0), analogously to the space ∧C which realizes a fundamental representation of SLnC. The Fock space has a natural basis indexed by certain infinite subsets of integers. The combinatorial part of our problem amounts to defining which of these subsets will index basis elements of Vw(Λ0) for a given w. We describe these special subsets in terms of a recursive but very simple algorithm, the roof operator on subsets. This is analogous to the left-key construction of Lascoux-Schutzenberger [11], which distinguishes the Young tableaux indexing a basis of a given Demazure module of SLnC. The roof operator is more elementary (and much more efficient) than the crystal graph operators, and is in some sense orthogonal to them. One may think of the roof operator as jumping across the crystal graph, moving each vertex down to an extremal weight vertex w(Λ0), but not along edges of the crystal graph. The combinatorics of the roof operator lead naturally to the definition of our standard basis, in analogy to the method of Raghavan-Sankaran [16]. A novel feature is that we define our basis “bottom-up” by raising each extremal weight
DOI：
--
发表时间：
2010
期刊：
影响因子：
--
作者：
Tanisaki;T.
通讯作者：
T.
DOI：
--
发表时间：
2005
期刊：
影响因子：
--
作者：
T. Kobayashi;T. Oshima
通讯作者：
T. Oshima