Mixed volumes of hypersimplices, root systems and shifted young tableaux
Mixed volumes of hypersimplices, root systems and shifted young tableaux
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混合体的超单纯形、根系统和移动的年轻画面
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发表时间：
2010
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通讯作者：
D. Croitoru
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作者：
D. Croitoru
本论文由两部分组成。在第一部分中，我们首先研究作为超单形的Minkowski和的经典置换面。它们的体积可以表示为多项式，其系数由超单形的混合体积给出。我们建立在Postnikov的结果，并推导出各种递归和组合公式的混合欧拉数。我们将这些结果推广到任意的根系4D，并获得循环，递归和组合公式的重量多面体（D-类似的置换面体）以及混合D-欧拉数的体积。这些公式涉及到Cartan矩阵和Dynkin图中的加权路径，从而使我们能够将混合欧拉数的理论扩展到其主子式可逆的任意矩阵。第二部分是对标准杨格变换造型中某些模式的研究。对于楼梯形状，Postnikov发现了由这些tableaux的对角条目和（标准）缔合面体的晶格点形成的向量之间的双射。使用类似的技术，我们推广到任意移动的形状这一结果。论文导师：亚历山大Postnikov职称：应用数学副教授
This thesis consists of two parts. In the first part, we start by investigating the classical permutohedra as Minkowski sums of the hypersimplices. Their volumes can be expressed as polynomials whose coefficients the mixed Eulerian numbers are given by the mixed volumes of the hypersimplices. We build upon results of Postnikov and derive various recursive and combinatorial formulas for the mixed Eulerian numbers. We generalize these results to arbitrary root systems 4D, and obtain cyclic, recursive and combinatorial formulas for the volumes of the weight polytopes (D-analogues of permutohedra) as well as the mixed D-Eulerian numbers. These formulas involve Cartan matrices and weighted paths in Dynkin diagrams, and thus enable us to extend the theory of mixed Eulerian numbers to arbitrary matrices whose principal minors are invertible. The second part deals with the study of certain patterns in standard Young tableaux of shifted shapes. For the staircase shape, Postnikov found a bijection between vectors formed by the diagonal entries of these tableaux and lattice points of the (standard) associahedron. Using similar techniques, we generalize this result to arbitrary shifted shapes. Thesis Supervisor: Alexander Postnikov Title: Associate Professor of Applied Mathematics