Half circles on flag manifolds over a semifield

TL;DR

This paper studies the structure of flag manifolds over semifields, showing that any two points can be connected through sequences of specific orbits called half i-circles, revealing new topological properties.

Contribution

It introduces the concept of half i-circles on flag manifolds over semifields and proves their connectedness properties for certain semifields.

Findings

Any two points in the flag manifold can be joined by a finite sequence of half i-circles.

The connectedness property holds for certain semifields.

The structure of flag manifolds over semifields is characterized by these orbit partitions.

Abstract

A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a connectedness property of the flag manifold: any two points of it can be joined by a finite sequence of half i-circles for various i.