On the Subdivision Algebra for the Polytope \(\mathcal {U}_{I,\overline{J}}\)

Pub Date : 2023-05-19 DOI:10.1007/s00026-023-00650-6
Matias von Bell, Martha Yip
{"title":"On the Subdivision Algebra for the Polytope \\(\\mathcal {U}_{I,\\overline{J}}\\)","authors":"Matias von Bell,&nbsp;Martha Yip","doi":"10.1007/s00026-023-00650-6","DOIUrl":null,"url":null,"abstract":"<div><p>The polytopes <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span> were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of <span>\\((I,\\overline{J})\\)</span>-Tamari lattices. They observed a connection between certain <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span> and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span>. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span> is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span> to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span>. As a consequence, this implies that subdivisions of <span>\\(\\mathcal {U}_{I,\\overline{J}}\\)</span> can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the <span>\\((I,\\overline{J})\\)</span>-Tamari complex can be obtained as a triangulated flow polytope.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00650-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

The polytopes \(\mathcal {U}_{I,\overline{J}}\) were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of \((I,\overline{J})\)-Tamari lattices. They observed a connection between certain \(\mathcal {U}_{I,\overline{J}}\) and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all \(\mathcal {U}_{I,\overline{J}}\). We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that \(\mathcal {U}_{I,\overline{J}}\) is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of \(\mathcal {U}_{I,\overline{J}}\) to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to \(\mathcal {U}_{I,\overline{J}}\). As a consequence, this implies that subdivisions of \(\mathcal {U}_{I,\overline{J}}\) can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the \((I,\overline{J})\)-Tamari complex can be obtained as a triangulated flow polytope.

Abstract Image

Abstract Image

分享
查看原文
关于多项式$$\mathcal的细分代数{U}_{I,\overline{J}}$$
多面体 \(\mathcal {U}_{I,\overline{J}}\) 是由 Ceballos、Padrol 和 Sarmiento 引入的,为研究 \((I,\overline{J})\)-Tamari 网格提供了一种几何方法。他们观察到了\(\mathcal {U}_{I,\overline{J}}) 和无环根多面体之间的联系,并想知道梅萨罗斯的细分代数是否可以用来细分所有的\(\mathcal {U}_{I,\overline{J}}) 。我们从两个角度对此做出了肯定的回答,一个是使用流多边形,另一个是使用根多边形。我们证明\(\mathcal {U}_{I,\overline{J}}\) 积分等价于可以用细分代数细分的流多胞形。或者,我们可以找到 \(\mathcal {U}_{I,\overline{J}}) 到无环根多面体的合适投影,它允许根多面体的细分被提升回 \(\mathcal {U}_{I,\overline{J}}) 。因此,这意味着 \(\mathcal {U}_{I,\overline{J}}\) 的细分可以用在细分代数中使用单项式的还原形式的代数解释来获得。此外,我们还证明了 \((I,\overline{J})\)-Tamari 复数可以作为三角流多面体得到。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信