Vol. 15, No. 8, 2021

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Flag manifolds over semifields

Huanchen Bao and Xuhua He

Vol. 15 (2021), No. 8, 2037–2069

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.

flag manifolds, Kac–Moody groups, total positivity
Mathematical Subject Classification
Primary: 14M15, 15B48, 20G44
Received: 8 July 2020
Revised: 24 November 2020
Accepted: 23 February 2021
Published: 10 November 2021
Huanchen Bao
Department of Mathematics
National University of Singapore
Xuhua He
The Institute of Mathematical Sciences and Department of Mathematics
Chinese University of Hong Kong
Shatin, N.T.
Hong Kong SAR