论文标题
通过避免图案，约翰逊图和静态图的阳性品种平滑度的标准
Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs
论文作者
论文摘要
阳性型是一定的代表性基质，最初是由Postnikov研究的，与完全非负晶石有关，现在在代数组合中广泛使用。阳性型产生确定性方程式，将阳性品种定义为格拉曼尼亚品种的亚变化。 Rietsch，Knutson-Lam-Speyer和Pawlowski研究了这些品种的几何和共同体学特性。在本文中，我们通过建立几种等效条件来继续研究阳性品种的几何特性，这些条件使用在装饰置换术中定义的模式回避的变化来表征光滑的阳性品种，这些变化是在装饰的排列中定义的，这些变体与阳性型阳性式的置换。这使我们可以给出两个公式，用于计算光滑阳性型的数量以及两个$ q $ -Analogs。此外，我们提供了一种组合方法，用于使用Johnson图的诱导子图在关键点确定阳性品种的切线空间的维度。我们还给出了阳性型的bruhat间隔表征。
Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. This allows us to give two formulas for counting the number of smooth positroids along with two $q$-analogs. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at key points using an induced subgraph of the Johnson graph. We also give a Bruhat interval characterization of positroids.
