Partial order on involutive permutations and double Schubert cells

TL;DR

This paper explores the partial order on involutive permutations arising from Borel subgroup actions and relates it to orbit closures in Grassmannian products, revealing a geometric connection between these contexts.

Contribution

It identifies and establishes a geometric relation between the partial order on involutions from orbit closures in different algebraic varieties.

Findings

The same partial order appears in two distinct geometric settings.

A geometric relation between orbit closures in upper-triangular matrices and Grassmannians is established.

The work links combinatorial involutions with geometric orbit structures.

Abstract

As shown by A. Melnikov, the orbits of a Borel subgroup acting by conjugation on upper-triangular matrices with square zero are indexed by involutions in the symmetric group. The inclusion relation among the orbit closures defines a partial order on involutions. We observe that the same order on involutive permutations also arises while describing the inclusion order on B-orbit closures in the direct product of two Grassmannians. We establish a geometric relation between these two settings.