Flag manifolds over semifields

TL;DR

This paper extends the theory of flag manifolds over semifields to Kac-Moody settings, establishing their structure, actions, cellular decompositions, and proving a Lusztig conjecture on duality, thus generalizing prior finite-type results.

Contribution

It introduces the theory of flag manifolds over semifields for Kac-Moody root data, including their cellular structure and monoid actions, extending finite-type results.

Findings

Flag manifolds over semifields admit cellular decompositions.

The monoid over the semifield acts naturally on the flag manifold.

Proved Lusztig's conjecture on duality of totally nonnegative flag manifolds.

Abstract

In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that the flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac-Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As a by-product, we prove a conjecture of Lusztig on the duality of totally nonnegative flag manifold of finite type.