On stationary real matrix Schubert varieties
Jaehoon Lee, Sangwoo Park, Eungbeom Yeon
TL;DR
The paper investigates when real matrix Schubert varieties are stationary (minimal) with respect to the first variation. It proves a necessary condition: if the partial permutation ω is non-vexillary, the open-dense part $X_{\omega}$ is not minimal, and it verifies minimality for a vexillary subclass with Grassmannian-type Rothe diagrams having at most two components (the set $\widetilde{\mathfrak{Gr}}_2$), using a normal-frame analysis and involutive isometries. It also extends minimality results to products and determinantal varieties, and provides a framework for decomposing certain varieties when the Rothe diagram contains the top-left corner. The methods combine explicit minor computations, Hessian-trace analysis in a constructed normal frame, and symmetry arguments to determine when the mean curvature vanishes, yielding new minimal cones and a deeper connection between combinatorial pattern avoidance and geometric minimality.
Abstract
In this paper, we study when a real matrix Schubert variety is stationary with respect to the first variation. We first show that a necessary condition for its open dense regular part to be a minimal submanifold is that the corresponding partial permutation is vexillary. Among vexillary partial permutations, we establish minimality by a geometric argument when the Rothe diagram is of Grassmannian type and has at most two connected components. We further obtain, as a corollary, the minimality of those varieties that decompose as products of this type. These varieties include all determinantal varieties as well as some new minimal cones.