The Path Model, the Quantum Frobenius Map and Standard Monomial Theory
The Path Model, the Quantum Frobenius Map and Standard Monomial Theory
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路径模型、量子弗罗贝尼乌斯图和标准单项式理论
DOI：
10.1007/978-94-011-5308-9_10
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
P. Littelmann
中科院分区：
文献类型：
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作者：
P. Littelmann
本文的目的是介绍表征路径模型理论及其相关基础。该理论的起点是一系列文章，其中 Lakshmibai、Musili 和 Seshadri 发起了一项计划，为具有一些特别好的几何性质的空间 H0(G/B,λ) 构建基础。这里我们假设 G 是一个在代数闭域 k 上定义的还原代数群，B 是一个固定的 Borel 子群，并且 λ 是与主导权重 λ 相关的标志变体 G/B 上的线丛。该程序的目的是将群 GL(n) 的 Hodge-Young 标准单项式理论扩展到任何半简单线性代数群的情况，更一般地说，扩展到 Kac-Moody 代数。除了这种构造的独立意义之外，这些结果对于表示组合学以及舒伯特簇的几何学也有重要的应用。对于几何应用，请注意，标准单项式理论提供了舒伯特簇上有效线丛的更高上同调的消失定理的证明，经典不变量理论中不变量环的显式基础，德马祖尔猜想的证明，舒伯特簇的正态性的证明，良好过滤性质的另一个证明，舒伯特簇的奇异轨迹的确定[9]，SL(n)/Binto a的变形复曲面品种[2]等
The aim of this article is to give an introduction to the theory of path models of representations and their associated bases. The starting point for the theory was a series of articles in which Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the spaceH0(G/B,λ) with some particularly nice geometric properties. Here we suppose thatGis a reductive algebraic group defined over an algebraically closed fieldk,Bis a fixed Borel subgroup, andλis the line bundle on the flag varietyG/Bassociated to a dominant weight λ. The purpose of the program is to extend the Hodge-Young standard monomial theory for the groupGL(n) to the case of any semisimple linear algebraic group and, more generally, to Kac-Moody algebras. Apart from the independent interest of such a construction, the results have important applications to the combinatorics of representations as well as to the geometry of Schubert varieties. For the geometric applications note that standard monomial theory provides proofs of the vanishing theorems for the higher cohomology of effective line bundles on Schubert varieties, explicit bases for the rings of invariants in classical invariant theory, a proof of Demazure’s conjecture, a proof of the normality of Schubert varieties, another proof of the good filtration property, a determination of the singular locus of Schubert varieties [9], a deformation ofSL(n)/Binto a toric variety [2], etc.
DOI：
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发表时间：
2007
期刊：
影响因子：
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作者：
T.Kobayashi;W.Schmid;and J.-H.Yang;editors
通讯作者：
editors