2026-09-18T20:11:27Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/824412022-01-13T07:54:04Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Richard P. Stanley.
Li, Nan, Ph. D. Massachusetts Institute of Technology
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2013-11-18T19:23:42Z
2013-11-18T19:23:42Z
2013
http://hdl.handle.net/1721.1/82441
862976320
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 69-70).
We studies two examples of polytope slices, hypersimplices as slices of hypercubes and edge polytopes. For hypersimplices, the main result is a proof of a conjecture by R. Stanley which gives an interpretation of the Ehrhart h*-vector in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes. We next study slices of edge polytopes. Let G be a finite connected simple graph with d vertices and let PG C Rd be the edge polytope of G. We call PG decomposable if PG decomposes into integral polytopes PG+ and PG- via a hyperplane, and we give an algorithm which determines the decomposability of an edge polytope. Based on a sequence of papers by Ohsugi and Hibi, we prove that when PG is decomposable, PG is normal if and only if both PG+ and PG- are normal. We also study toric ideals of PG, PG+ and PG-. This part is joint work with Hibi and Zhang.
by Nan Li.
Ph.D.
70 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Combinatorial aspects of polytope slices
Thesis
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Publication
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<Abstract>We studies two examples of polytope slices, hypersimplices as slices of hypercubes and edge polytopes. For hypersimplices, the main result is a proof of a conjecture by R. Stanley which gives an interpretation of the Ehrhart h*-vector in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes. We next study slices of edge polytopes. Let G be a finite connected simple graph with d vertices and let PG C Rd be the edge polytope of G. We call PG decomposable if PG decomposes into integral polytopes PG+ and PG- via a hyperplane, and we give an algorithm which determines the decomposability of an edge polytope. Based on a sequence of papers by Ohsugi and Hibi, we prove that when PG is decomposable, PG is normal if and only if both PG+ and PG- are normal. We also study toric ideals of PG, PG+ and PG-. This part is joint work with Hibi and Zhang.</Abstract>
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