Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras
Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras
复制标题
可对称 Kac-Moody 代数的收缩模和标准单项式理论
DOI：
10.1090/s0894-0347-98-00268-9
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发表时间：
1998
影响因子：
3.9
通讯作者：
P. Littelmann
中科院分区：
文献类型：
--
作者：
P. Littelmann
设G是定义在代数闭域k上的约化代数群，我们固定一个Borel子群B，对于一个支配权λ，设Lλ是广义标志簇G/B上的相关线丛。在一系列文章中，Lakshmibai，Musili和Seshadri开始了一个程序，构造了具有一些特别好的几何性质的空间H0(G/B，Lλ)的基。该程序的目的是将群SL(N)的Hodge-Young标准单项式理论推广到任何半单代数群的情形，更一般地，推广到Kac-Moody代数。我们参考文献[3]、[7]、[10]、[14]对该主题和应用进行调查。我们提供了一种新的方法，它完成了程序，避免了前面文章中逐个案例的考虑。事实上，该方法适用于所有可对称化的Kac-Moody代数。在我们的方法中，我们需要的最重要的工具是表示[11]、[12]的路径模型的组合语言，以及位于单位根的量子群。设UV(G)是与G有单位根v的量子群。我们利用量子Frobenius映射[15]来“压缩”某些UV(G)-模，使它们成为G-模。对偶空间之间的对应映射可以看作是幂映射H(G/B，Lλ)→H(G/B，L‘λ)，S 7→S的一种分裂。为了简单起见，我们假设我们处于简单的花边情况。设Vλ是G的最高权λ的Weyl模，Mλ是UV(G)的相应Weyl模。张量积bπ：=bν1⊗有一种典型的附加方法。。.⊗bν‘of Extreal Weight向量bνj∈M∗λ到形状为π[11]的每个L-S路径λ[11]以获得适当的`(回想L-S路径可以由极值权重和有理数的集合来表征)。为了构造H0(G/B，Lλ)=V∗λ的基，我们利用压缩映射将Vλ嵌入到(Mλ)⊗`.用pπ表示对偶映射(MπV∗λ.)下的b∗λ)⊗`→在V∗λ中的像我们证明了向量pπ，π和L-S路径的形状λ，构成了V∗λ的基。此外，`次方pπ∈H0(G/B，L`λ)是极权向量pν1···pν`，pνi∈H0(G/B，Lλ)的乘积，加上在某些偏序中较大的元素的线性组合。P-π给出的基与Schubert簇X的限制映射H0(G/B，Lλ)→H0(X，Lλ))相容，且具有“标准单项式性质”。
Let G be a reductive algebraic group defined over an algebraically closed field k. We fix a Borel subgroup B, and for a dominant weight λ let Lλ be the associated line bundle on the generalized flag variety G/B. In a series of articles, Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H0(G/B,Lλ) with some particularly nice geometric properties. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group SL(n) to the case of any semisimple algebraic group and, more generally, to Kac-Moody algebras. We refer to [3], [7], [10], [14] for a survey of the subject and applications. We provide a new approach which completes the program and which avoids the case by case considerations of the earlier articles. In fact, the method works for all symmetrizable Kac-Moody algebras. The most important tools we need in our approach are the combinatorial language of the path model of a representation [11], [12], and quantum groups at a root of unity. Let Uv(g) be the quantum group associated to G at an `-th root of unity v. We use the quantum Frobenius map [15] to “contract” certain Uv(g)-modules so that they become G-modules. The corresponding map between the dual spaces can be seen as a kind of splitting of the power map H(G/B,Lλ) → H(G/B,L`λ), s 7→ s. For simplicity let us assume we are in the simply laced case. Let Vλ be the Weyl module of G of highest weight λ, and let Mλ be the corresponding Weyl module of Uv(g). There is a canonical way to attach a tensor product bπ := bν1 ⊗ . . .⊗ bν` of extremal weight vectors bνj ∈ M∗ λ to each L-S path π of shape λ [11] for an appropriate ` (recall that an L-S path can be characterized by a collection of extremal weights and rational numbers). To construct a basis of H0(G/B,Lλ) = V ∗ λ , we use the contraction map to embed Vλ into (Mλ) ⊗`. Denote by pπ the image of bπ in V ∗ λ under the dual map (M ∗ λ) ⊗` → V ∗ λ . We show that the vectors pπ, π an L-S path of shape λ, form a basis of V ∗ λ . Further, the `-th power pπ ∈ H0(G/B,L`λ) is a product of extremal weight vectors pν1 · · · pν` , pνi ∈ H0(G/B,Lλ), plus a linear combination of elements which are “bigger” in some partial order. The basis given by the pπ is compatible with the restriction map H0(G/B,Lλ) → H0(X,Lλ) to a Schubert variety X , and it has the “standard monomial property”.