{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/polyhedral-and-tropical-geometry-of-flag","title":"Polyhedral and Tropical Geometry of Flag Positroids","arxiv_id":"2208.09131","date":"2022-08-19","proceeding":null,"authors":["Jonathan Boretsky","Christopher Eur","Lauren Williams"],"abstract":"A flag positroid of ranks $\\boldsymbol{r}:=(r_1<\\dots <r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k \\times n$ matrix $A$ such that the $r_i \\times r_i$ minors of $A$ involving rows $1,2,\\dots,r_i$ are nonnegative for all $1\\leq i \\leq k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $\\boldsymbol{r}:=(a, a+1,\\dots,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl$_{\\boldsymbol{r},n}^{\\geq 0}$ equals the nonnegative flag Dressian FlDr$_{\\boldsymbol{r},n}^{\\geq 0}$, and that the points $\\boldsymbol{\\mu} = (\\mu_a,\\ldots, \\mu_b)$ of TrFl$_{\\boldsymbol{r},n}^{\\geq 0} =$ FlDr$_{\\boldsymbol{r},n}^{\\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\\underline{\\boldsymbol{\\mu}})$ into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its $(\\leq 2)$-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids $(\\chi_1,\\dots,\\chi_k)$ which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks $\\boldsymbol{r}=(a,a+1,\\dots,b)$ is realizable.","url_abs":"https://arxiv.org/abs/2208.09131v5","url_pdf":"https://arxiv.org/pdf/2208.09131v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"polyhedral-and-tropical-geometry-of-flag","repo_url":"https://github.com/chrisweur/postropflagvar","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}