摘要翻译：
本文给出了如何计算正交Grassmannian中Schubert簇点上的多重数，或者更一般的Hilbert函数的解。应用标准单体理论将问题从几何学转化为组合学。对由此产生的组合问题的解决构成了论文的大部分。这种方法以前已经被用来解决Grassmannian和辛Grassmannian的相同问题。作为应用，我们将多重性解释为一类非相交格路径的数目。把舒伯特变体看作是一种特殊的变体，把点看作是“同一陪集”，我们的问题专门讨论关于Pfaffian理想处理的问题，在文献中有不同的方法对它进行处理。当点是“一般奇点”时，文献中也有一个几何解。
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英文标题：
《Hilbert functions of points on Schubert varieties in Orthogonal
  Grassmannians》
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作者：
K. N. Raghavan and Shyamashree Upadhyay
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian.   As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind.   Taking the Schubert variety to be of a special kind and the point to be the "identity coset," our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a "generic singularity."
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PDF链接：
https://arxiv.org/pdf/0704.0542