Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties (Xw̄) and Kazhdan-Lusztig varieties (Nv,w). Matrix Schubert varieties can be written as Xw̄ = Yw × Cᵈ, where d is maximal. We are especially interested in the structure and complexity of these varieties Yw and Nv,w under the so-called usual torus actions. In the case when Yw is toric, we provide a full characterisation of the simple reflections sᵢ that render Yw · sᵢ 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.