Title: Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models Show Chinese title

Title: 环面在矩阵Schubert和Kazhdan-Lusztig簇上的作用及其与统计模型的联系

Abstract: We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties can be written as $\overline{X_w} = Y_w \times \mathbb C^d$, where $d$ is maximal. We are especially interested in the structure and complexity of these varieties $Y_w$ and $\mathcal{N}_{v,w}$ under the so-called usual torus actions. In the case when $Y_w$ is toric, we provide a full characterisation of the simple reflections $s_i$ that render ${Y_{w \cdot s_i}}$ toric, as well as the corresponding changes to the weight cone. For Kazhdan-Lusztig varieties, we consider how moving one of the two permutations $v,w$ along a chain in the Bruhat poset affects their complexity. Additionally, we study the complexity of these varieties, for permutations $v$ and $w$ of a specific structure. Finally, we consider the links between these determinantal varieties and two classes of statistical models; namely conditional independence and quasi-independence models. Show Chinese abstract

Abstract: 我们研究从排列中产生的两类广义行列式簇的环面几何：矩阵Schubert簇（$\overline{X_w}$）和Kazhdan-Lusztig簇（$\mathcal{N}_{v,w}$）。矩阵Schubert簇可以写成$\overline{X_w} = Y_w \times \mathbb C^d$，其中$d$是最大的。我们特别关注这些簇$Y_w$和$\mathcal{N}_{v,w}$在所谓的通常环面作用下的结构和复杂性。 当$Y_w$为扇形时，我们提供了对简单反射$s_i$的完整特征描述，这些反射使${Y_{w \cdot s_i}}$为扇形，以及相应的权锥的变化。 对于 Kazhdan-Lusztig 变体，我们考虑在 Bruhat 偏序集中的链上移动两个排列之一$v,w$对其复杂性的影响。 此外，我们研究了这些变体的复杂性，针对具有特定结构的排列$v$和$w$。 最后，我们考虑这些行列式变体与两类统计模型之间的联系；即条件独立性和准独立性模型。