Flag manifolds over semifields

Bao, Huanchen; He, Xuhua

doi:10.2140/ant.2021.15.2037

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Flag manifolds over semifields

Huanchen Bao and Xuhua He

Vol. 15 (2021), No. 8, 2037–2069

DOI: 10.2140/ant.2021.15.2037

Abstract

In this paper, we develop the theory of flag manifolds over a semifield for any Kac–Moody root datum. We show that a 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 an important consequence, we prove a conjecture of Lusztig on the duality of a totally nonnegative flag manifold of finite type.

Keywords

flag manifolds, Kac–Moody groups, total positivity

Mathematical Subject Classification

Primary: 14M15, 15B48, 20G44

Milestones

Received: 8 July 2020

Revised: 24 November 2020

Accepted: 23 February 2021

Published: 10 November 2021

Authors

Huanchen Bao
Department of Mathematics National University of Singapore Singapore

Xuhua He
The Institute of Mathematical Sciences and Department of Mathematics Chinese University of Hong Kong Shatin, N.T. Hong Kong SAR China

Vol. 15, No. 8, 2021