Mixed volumes of hypersimplices, root systems and shifted young tableaux
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2011
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| Online Access: | http://hdl.handle.net/1721.1/64610 |
| _version_ | 1826207245319274496 |
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| author | Croitoru, Dorian (Dorian Eugen) |
| author2 | Alexander Postnikov. |
| author_facet | Alexander Postnikov. Croitoru, Dorian (Dorian Eugen) |
| author_sort | Croitoru, Dorian (Dorian Eugen) |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. |
| first_indexed | 2024-09-23T13:46:38Z |
| format | Thesis |
| id | mit-1721.1/64610 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T13:46:38Z |
| publishDate | 2011 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/646102019-04-10T18:04:57Z Mixed volumes of hypersimplices, root systems and shifted young tableaux Croitoru, Dorian (Dorian Eugen) Alexander Postnikov. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. Cataloged from PDF version of thesis. Includes bibliographical references (p. 40-41). 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. by Dorian Croitoru. Ph.D. 2011-06-20T15:59:55Z 2011-06-20T15:59:55Z 2010 2010 Thesis http://hdl.handle.net/1721.1/64610 727162608 eng 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 41 p. application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Croitoru, Dorian (Dorian Eugen) Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title | Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title_full | Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title_fullStr | Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title_full_unstemmed | Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title_short | Mixed volumes of hypersimplices, root systems and shifted young tableaux |
| title_sort | mixed volumes of hypersimplices root systems and shifted young tableaux |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/64610 |
| work_keys_str_mv | AT croitorudoriandorianeugen mixedvolumesofhypersimplicesrootsystemsandshiftedyoungtableaux |