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

TL;DR

The paper investigates the usual torus action on two families of determinantal varieties—matrix Schubert varieties and Kazhdan-Lusztig varieties—within the framework of T-varieties, and develops concrete combinatorial tools (opposite Rothe diagrams, edge cones, and associated DAGs) to compute weight cones and complexity. It provides a complete combinatorial characterization of when a toric matrix Schubert piece $Y_w$ remains toric after right-multiplication by a simple reflection, and analogous analyses for KL varieties, including how toric properties behave under Bruhat-interval glueing and extensions. A central contribution is linking the geometry to statistics by identifying CI and quasi-independence models realized by these varieties, establishing complexity results, and proving that toric quasi-independence models have rational MLE via graph-theoretic criteria. The results unite algebraic geometry, combinatorics, and algebraic statistics, offering explicit criteria, dimension counts, and constructive methods (with Macaulay2 packages) to study orbit closures, Bruhat intervals, and statistical models arising from determinantal ideals.

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.

Figures (3)

• Figure 1: The figure on the left depicts the hook in an $n\times n$ lattice. The figure on the right illustrates the shapes of $L(w), L'(w)$ and the dominant piece, which is drawn by the grey area.
• Figure 2: The image on the left depicts the form of the opposite Rothe diagram in the case of multiple disjoint hooks. The grey area is the dominant piece. In the diagram on the right, we have two steps $s^i$ and $s^{i+1}$. The ones in blue lie on the step, since their position on the step is lower than the height of the step. The one in pink lies north of the step, since that is not the case for it.
• Figure 3: One can see the structure of $v$ and the structure of $w$ in each of the cases. The area of $v$ that is outlined in green corresponds to the covariance matrix $\Sigma$. The diagonal of $\Sigma$ is highlighted in pink. In both cases, the Fulton conditions coming from $w$ are the same as the conditions from the corresponding CI statement, which are highlighted in the same colour in $\Sigma$.

Theorems & Definitions (106)

• Theorem 1.1: Theorem \ref{['thm: all toric mS']}, Corollary \ref{['cor: toric cases']}, \ref{['cor: change of the weight cone']}
• Theorem 1.2: Lemma \ref{['lem: connected components for no unexpected zeros']}, \ref{['lem: isolated vertices for no unexpected zeros']}
• Definition 1.3
• Lemma 1.4: donten2021complexity
• Example 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Theorem 2.5: fulton1992flags
• Example 2.6