{"title":"On the Subdivision Algebra for the Polytope \\(\\mathcal {U}_{I,\\overline{J}}\\)","authors":"Matias von Bell, 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.