Total positivity in twisted product of flag varieties
Abstract
We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit parameterizations of each totally positive cell. In the special cases of double flag varieties and braid varieties, we show that the totally nonnegative parts are regular CW complexes homeomorphic to closed balls. Moreover, we prove that the link of any totally nonnegative double Bruhat cell in a reductive group is a regular CW complex homeomorphic to a closed ball, solving an open problem of Fomin and Zelevinsky.
Keywords
Cite
@article{arxiv.2211.11168,
title = {Total positivity in twisted product of flag varieties},
author = {Huanchen Bao and Xuhua He},
journal= {arXiv preprint arXiv:2211.11168},
year = {2024}
}
Comments
v2, 21 pages. The previous proof of shelling in Theorem 4.1 in v1 was incorrect, hence the regularity of general twisted products. In v2, we prove the shelling in the setting of double flag varieties. Regularity results remain valid for double flag varieties, links on double Bruhat cells, and braid varieties. The abstract is updated