论文标题
标志阳性型的多面体和热带几何形状
Polyhedral and Tropical Geometry of Flag Positroids
论文作者
论文摘要
$ [n] $上的等级$ \ boldsymbol {r}的标志阳性型：=（r_1 <\ dots <r_k）$是一个flag r_k \ times n $ matrix $ a $可以实现的标志Matroid $ 1 \ leq i \ leq k $。在本文中，我们探讨了标志阳性型的多面体和热带几何形状，尤其是当$ \ boldsymbol {r}：=（a，a+1，\ dots，b）$是连续数字的顺序。在这种情况下，我们表明非负热带标志品种trfl $ _ {\ boldsymbol {r}，n}^{\ geq 0} $等于非负flag fressian fldr $ _ {\ boldsymbol {\ boldsymbol {r} （μ_a，\ ldots，μ_b）trfl $ _ {\ boldsymbol {r}，n}^{\ geq 0} = $ fldr $ _ {\ boldsymbol {r}，n} $ p（\ suespline {\boldsymbolμ}）$在positroid polytopes中。我们的结果应用于Bruhat间隔多型：例如，当且仅当其$（\ leq 2）$ - 尺寸面是bruhat间隔的polytopes时，我们表明完整的标志Matroid polytope是Bruhat间隔多层。我们的结果还适用于可靠性问题。我们将正向的标志矩阵定义为一系列取向的矩阵$（χ_1，\ dots，χ_k）$，它也是一个方向的标志矩阵。然后，我们证明，每个等级的标准标志$ \ boldsymbol {r} =（a，a，a+1，\ dots，b）$都是可实现的。
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 $\boldsymbolμ = (μ_a,\ldots, μ_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$ FlDr$_{\boldsymbol{r},n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\boldsymbolμ})$ 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 $(χ_1,\dots,χ_k)$ 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.
