Product structure and regularity theorem for totally nonnegative flag varieties

TL;DR

This paper introduces a new form of total positivity called J-total positivity for Kac-Moody groups, establishing cellular decompositions and topological regularity results for totally nonnegative flag varieties, confirming several conjectures.

Contribution

It generalizes total positivity to Kac-Moody groups, constructs cellular decompositions, and proves topological regularity of closures of positive Richardson varieties.

Findings

Cellular decomposition of J-totally nonnegative flag varieties

Regular CW complex homeomorphic to a closed ball for each positive Richardson variety

Validation of conjectures by Galashin, Karp, and Lam

Abstract

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.