On stationary real matrix Schubert varieties

TL;DR

This paper characterizes when real matrix Schubert varieties are stationary by linking minimality to vexillary permutations and geometric conditions, identifying new minimal cones and extending known results.

Contribution

It establishes minimality conditions for real matrix Schubert varieties based on vexillary permutations and geometric properties, including new classes of minimal cones.

Findings

Vexillary partial permutations are necessary for minimality.

Minimality is proven for varieties with Grassmannian-type Rothe diagrams with up to two components.

Includes new minimal cones and extends minimality to product varieties.

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.