Two partitions of a flag manifold

TL;DR

This paper explores two different partitions of a flag manifold, one based on Frobenius action and the other on conjugation by a regular semisimple element, highlighting their similarities.

Contribution

It identifies and analyzes the similarities between two distinct partitions of a flag manifold, connecting Frobenius action and conjugation by regular semisimple elements.

Findings

Both partitions are indexed by Weyl group elements.

The partitions exhibit structural similarities.

The work bridges algebraic and geometric perspectives on flag manifolds.

Abstract

We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the other partition is defined using conjugation by a regular semisimple element.