Schubert polynomials and Bott-Samelson varieties
Schubert polynomials and Bott-Samelson varieties
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舒伯特多项式和 Bott-Samelson 簇
DOI：
10.1007/s000140050071
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发表时间：
1998
影响因子：
0.9
通讯作者：
P. Magyar
中科院分区：
文献类型：
--
作者：
P. Magyar
摘要。舒伯特多项式对舒尔多项式进行了一般化，但如何一般化几个经典公式：Weyl特征公式、Demazure特征公式和半标准表的生成级数，目前还不清楚。我们得到了这些缺失的公式，并得到了舒伯特多项式的几个令人惊奇的表达式。上面的结果自然地来自于一个新的基于bot - samelson变量的Schubert多项式几何模型。我们的分析包括了一个新的，明确的结构，一个bot - samelson品种Z作为一个b轨道的闭包在一个标志品种的产品。这种构造适用于任意约化群G，并且对于G = GL(n)实现了Z作为某偏序集合的表示。这个序集统一了几个著名的组合结构：广义杨图及其相关的舒尔模块；减少排列分解；以及Berenstein-Fomin-Zelevinsky的室集，它们在正则基和矩阵分解的组合学中至关重要。另一方面，我们对Z的嵌入给出了其坐标环的初等构造，并允许我们指定一个由tableaux索引的基。
Abstract. Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.¶The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott-Samelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.¶This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux.