Total positivity in twisted product of flag varieties

TL;DR

This paper proves that the totally nonnegative parts of twisted products of flag varieties have cellular decompositions, are topological manifolds with boundary, and in special cases are homeomorphic to closed balls, solving an open problem.

Contribution

It establishes cellular decompositions and explicit parameterizations of totally positive cells in twisted products of flag varieties, including special cases and an open problem.

Findings

Totally nonnegative parts admit cellular decompositions.

Each cell's closure is a topological manifold with boundary.

In special cases, they are homeomorphic to closed balls.

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.