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Mixed volumes of hypersimplices, root systems and shifted young tableaux

Author(s)
Croitoru, Dorian (Dorian Eugen)
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Massachusetts Institute of Technology. Dept. of Mathematics.
Advisor
Alexander Postnikov.
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M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582
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Abstract
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 [fee], and obtain cyclic, recursive and combinatorial formulas for the volumes of the weight polytopes ([fee]-analogues of permutohedra) as well as the mixed [fee]-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.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
 
Cataloged from PDF version of thesis.
 
Includes bibliographical references (p. 40-41).
 
Date issued
2010
URI
http://hdl.handle.net/1721.1/64610
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Massachusetts Institute of Technology
Keywords
Mathematics.

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