$B_{n-1}$-orbits on the flag variety and the Bruhat graph of the symmetric group

TL;DR

This paper provides a new combinatorial framework for understanding $B_{n-1}$-orbits on the flag variety of $GL(n)$, linking orbit closures to Bruhat order and Weyl group actions, and offers a simple formula for orbit dimensions.

Contribution

It introduces a novel combinatorial description of $B_{n-1}$-orbits via pairs of Weyl group elements and clarifies their closure relations using Bruhat order, improving computational methods.

Findings

Orbit closure relations are described via Bruhat order.

A simple formula for the dimension of each orbit is provided.

The approach enhances computational efficiency of orbit analysis.

Abstract

Let $G=G_{n}=GL(n)$ be the $n\times n$ complex general linear group and embed $G_{n-1}=GL(n-1)$ in the top left hand corner of $G$. The standard Borel subgroup of upper triangular matrices $B_{n-1}$ of $G_{n-1}$ acts on the flag variety of $G$ with finitely many orbits. In this paper, we show that each $B_{n-1}$-orbit is the intersection of orbits of two Borel subgroups of $G$ acting on the flag variety of $G$. This allows us to give a new combinatorial description of the $B_{n-1}$-orbits by associating to each orbit a pair of Weyl group elements. The closure relations for the $B_{n-1}$-orbits can then be understood in terms of the Bruhat order on the Weyl group, and the Richardson-Springer monoid action on the orbits can be understood in terms of the classical monoid action of the Weyl group on itself. This approach makes the closure relation more transparent than in earlier work of Magyar and the monoid action significantly more computable than in our earlier papers, and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit.