Hypersimplicial subdivisions

Let π:Rn→Rd be any linear projection, let A be the image of the standard basis. Motivated by Postnikov’s study of postitive Grassmannians via plabic graphs and Galashin’s connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions...

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Autores: Olarte, Jorge Alberto, Santos, Francisco|||0000-0003-2120-9068
Tipo de documento: artigo
Data de publicação:2022
País:España
Recursos:Universidad de Cantabria (UC)
Repositório:UCrea Repositorio Abierto de la Universidad de Cantabria
Idioma:inglês
OAI Identifier:oai:repositorio.unican.es:10902/27752
Acesso em linha:https://hdl.handle.net/10902/27752
Access Level:Acceso aberto
Palavra-chave:Hypersimplex
Subdivisions
Fiber polytope
Baues problem
Separated sets
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spelling Hypersimplicial subdivisionsOlarte, Jorge AlbertoSantos, Francisco|||0000-0003-2120-9068HypersimplexSubdivisionsFiber polytopeBaues problemSeparated setsLet π:Rn→Rd be any linear projection, let A be the image of the standard basis. Motivated by Postnikov’s study of postitive Grassmannians via plabic graphs and Galashin’s connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of π to the k-th hypersimplex, for k=1,…,n−1 . We show that: For arbitrary A and for k≤d+1 , the corresponding fiber polytope F(k)(A) is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of A of size max{d+2,n−k+1} . When A=Pn is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of π -coherent subdivisions into the poset of all π -induced subdivisions is a homotopy equivalence. When A=C(d,n) is the vertex set of a cyclic d-polytope with d odd and any n≥d+3, there are non-lifting (and even more so, non-separated) π -induced subdivisions for k=2.The authors were supported by the Einstein Foundation Berlin under grant EVF-2015-230. Work of F. Santos is also supported by grants MTM2017-83750-P/AEI/10.13039/501100011033 and PID2019-106188GB-I00/AEI/10.13039/501100011033 of the Spanish State Research Agency.SpringerUniversidad de Cantabria20222022-01-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttps://hdl.handle.net/10902/27752Selecta Mathematica, New Series, 2022, 28, 4reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/277522026-06-02T12:39:31Z
dc.title.none.fl_str_mv Hypersimplicial subdivisions
title Hypersimplicial subdivisions
spellingShingle Hypersimplicial subdivisions
Olarte, Jorge Alberto
Hypersimplex
Subdivisions
Fiber polytope
Baues problem
Separated sets
title_short Hypersimplicial subdivisions
title_full Hypersimplicial subdivisions
title_fullStr Hypersimplicial subdivisions
title_full_unstemmed Hypersimplicial subdivisions
title_sort Hypersimplicial subdivisions
dc.creator.none.fl_str_mv Olarte, Jorge Alberto
Santos, Francisco|||0000-0003-2120-9068
author Olarte, Jorge Alberto
author_facet Olarte, Jorge Alberto
Santos, Francisco|||0000-0003-2120-9068
author_role author
author2 Santos, Francisco|||0000-0003-2120-9068
author2_role author
dc.contributor.none.fl_str_mv Universidad de Cantabria
dc.subject.none.fl_str_mv Hypersimplex
Subdivisions
Fiber polytope
Baues problem
Separated sets
topic Hypersimplex
Subdivisions
Fiber polytope
Baues problem
Separated sets
description Let π:Rn→Rd be any linear projection, let A be the image of the standard basis. Motivated by Postnikov’s study of postitive Grassmannians via plabic graphs and Galashin’s connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of π to the k-th hypersimplex, for k=1,…,n−1 . We show that: For arbitrary A and for k≤d+1 , the corresponding fiber polytope F(k)(A) is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of A of size max{d+2,n−k+1} . When A=Pn is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of π -coherent subdivisions into the poset of all π -induced subdivisions is a homotopy equivalence. When A=C(d,n) is the vertex set of a cyclic d-polytope with d odd and any n≥d+3, there are non-lifting (and even more so, non-separated) π -induced subdivisions for k=2.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/10902/27752
url https://hdl.handle.net/10902/27752
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv Selecta Mathematica, New Series, 2022, 28, 4
reponame:UCrea Repositorio Abierto de la Universidad de Cantabria
instname:Universidad de Cantabria (UC)
instname_str Universidad de Cantabria (UC)
reponame_str UCrea Repositorio Abierto de la Universidad de Cantabria
collection UCrea Repositorio Abierto de la Universidad de Cantabria
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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