{
  "id": "2208.09131",
  "version": "v1",
  "published": "2022-08-19T03:05:09.000Z",
  "updated": "2022-08-19T03:05:09.000Z",
  "title": "Polyhedral and Tropical Geometry of Flag Positroids",
  "authors": [
    "Jonathan Boretsky",
    "Christopher Eur",
    "Lauren Williams"
  ],
  "comment": "40 pages",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-19T03:05:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tropical geometry",
      "bruhat interval polytope",
      "tropical flag variety trfl",
      "positively oriented flag matroid",
      "flag positroid polytope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}