Total positivity in twisted flag varieties

TL;DR

This work extends total positivity from ordinary flag varieties to twisted J‑flag varieties for Kac–Moody groups, proving that the totally nonnegative part $\mathcal{B}^J_{\ge0}$ admits a cell decomposition with closures that are regular CW complexes. The authors develop a twisted product structure and a Marsh–Rietsch–style parametrization suitable for the twisted setting, establishing both the nonempty pattern and the dimension formula $\dim\mathring{\mathcal{B}}^J_{v,w}=\ell^J(w)-\ell^J(v)$, and they show $\mathcal{B}^J_{\ge0}$ is a remarkable polyhedral space with regularity. They further extend these constructions to the totally nonnegative double flag variety via a thickening method, obtaining analogous CW-regularity and shellability results for $\mathcal{Z}_{\ge0}$, and they relate the positivity theory to representation-theoretic objects such as canonical bases in tensor products. The paper also proves that certain links in totally nonnegative double Bruhat cells are regular CW complexes, answering questions raised by Fomin–Zelevinsky and generalizing Hersh’s results. Altogether, the results provide a unified, combinatorially controlled framework for total positivity in twisted flag varieties and their doubles, with implications for canonical bases and higher Teichmüller-type structures.

Abstract

Let $G$ be a Kac-Moody group, split over $\mathbb R$. The totally nonnegative part of $G$ and its (ordinary) flag variety $G/B^+$ was introduced by Lusztig. It is known that the totally nonnegative parts of $G$ and $G/B^+$ have remarkable combinatorial and topological properties. In this paper, we consider the totally nonnegative part of the $J$-twisted flag variety $G/{}^J B^+$, where ${}^J B^+$ is the Borel subgroup opposite to $B^+$ in the standard parabolic subgroup $P_J^+$ of $G$. The $J$-twisted flag varieties include the ordinary flag variety $G/B^+$ as a special case. Our main result show that the totally nonnegative part of $G/{}^J B^+$ decomposes into cells, and the closure of each cell is a regular CW complex. This generalizes the work of Galashin-Karp-Lam \cite{GKL22} and the joint work of Bao with the first author \cite{BH24} for ordinary flag varieties. As an application, we deduce that the totally nonnegative part of the double flag variety $G/B^+ \times G/B^-$ with respect to the diagonal $G$-action has similar nice properties. We also establish some connections between the totally nonnegative part of the double flag with the canonical basis of the tensor product of a lowest weight module with a highest weight module of $G$. As another application, we show that the link of identity in a totally nonnegative reduced double Bruhat cell of $G$ is a regular CW complex. This generalizes the work of Hersh \cite{Her14} on the link of $U_{\geq0}^-$ and gives a positive answer to an open question of Fomin and Zelevinsky.

Theorems & Definitions (98)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Example 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4