{"title":"Polypositroids","authors":"Thomas Lam, Alexander Postnikov","doi":"10.1017/fms.2024.11","DOIUrl":null,"url":null,"abstract":"<p>We initiate the study of a class of polytopes, which we coin <span>polypositroids</span>, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315142604314-0934:S2050509424000112:S2050509424000112_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(W,c)$</span></span></img></span></span><span>-polypositroid</span> for a finite Weyl group <span>W</span> and a choice of Coxeter element <span>c</span>. We connect the theory of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315142604314-0934:S2050509424000112:S2050509424000112_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(W,c)$</span></span></img></span></span>-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss <span>membranes</span>, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Sigma","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fms.2024.11","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.
我们开始研究一类多面体，并将其定义为多正多面体（polypositroids），即同时是广义多面体（或多矩阵）和椭圆多面体的多面体。正多面体是由完全非负的格拉斯曼矩阵产生的矩阵，而多正多面体则是 "正 "多面体。我们用考克斯特项链和平衡图对多正多面体进行参数化，并用极值射线和面不等式描述多正多面体的锥面。我们引入了有限韦尔群 W 和所选 Coxeter 元素 c 的 $(W,c)$-多正多面体的概念。我们讨论了多正多面体内部的某些三角形二维曲面--膜。膜的概念从正多面体扩展到了多正多面体。
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Forum of Mathematics, Sigma is the open access alternative to the leading specialist mathematics journals. Editorial decisions are made by dedicated clusters of editors concentrated in the following areas: foundations of mathematics, discrete mathematics, algebra, number theory, algebraic and complex geometry, differential geometry and geometric analysis, topology, analysis, probability, differential equations, computational mathematics, applied analysis, mathematical physics, and theoretical computer science. This classification exists to aid the peer review process. Contributions which do not neatly fit within these categories are still welcome.
Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas will be welcomed. All published papers will be free online to readers in perpetuity.
$(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of 
$(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.
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