Simplex and Polygon Equations

It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a famil...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2015
Main Authors: Dimakis, A., Müller-Hoissen, F.
Format: Article
Language:English
Published: Інститут математики НАН України 2015
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/147105
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Simplex and Polygon Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 107 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Dimakis, A.
Müller-Hoissen, F.
author_facet Dimakis, A.
Müller-Hoissen, F.
citation_txt Simplex and Polygon Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 107 назв. — англ.
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of ''polygon equations'' realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N+1)-gon equation, its dual, and a compatibility equation.
first_indexed 2025-11-27T20:19:10Z
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institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-11-27T20:19:10Z
publishDate 2015
publisher Інститут математики НАН України
record_format dspace
spelling Dimakis, A.
Müller-Hoissen, F.
2019-02-13T16:25:31Z
2019-02-13T16:25:31Z
2015
Simplex and Polygon Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 107 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 06A06; 06A07; 52Bxx; 82B23
DOI:10.3842/SIGMA.2015.042
https://nasplib.isofts.kiev.ua/handle/123456789/147105
It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of ''polygon equations'' realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N+1)-gon equation, its dual, and a compatibility equation.
We have to thank an anonymous referee for comments that led to some corrections in our
 previous version of Section 2.2.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Simplex and Polygon Equations
Article
published earlier
spellingShingle Simplex and Polygon Equations
Dimakis, A.
Müller-Hoissen, F.
title Simplex and Polygon Equations
title_full Simplex and Polygon Equations
title_fullStr Simplex and Polygon Equations
title_full_unstemmed Simplex and Polygon Equations
title_short Simplex and Polygon Equations
title_sort simplex and polygon equations
url https://nasplib.isofts.kiev.ua/handle/123456789/147105
work_keys_str_mv AT dimakisa simplexandpolygonequations
AT mullerhoissenf simplexandpolygonequations