Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties

TL;DR

This paper explores the structure and dynamics of totally nonnegative parts of flag varieties viewed as adjoint orbits, introducing new maps, analyzing gradient flows, and establishing topological properties of related geometric objects.

Contribution

It introduces the totally nonnegative orbit framework, defines a generalized twist map, and analyzes positivity-preserving gradient flows across different metrics.

Findings

The totally nonnegative part of orbits is explicitly described in several cases.

Positivity is preserved under the Kähler gradient flow in many cases.

Twisted Vandermonde amplituhedra are homeomorphic to closed balls.

Abstract

One can view a partial flag variety in $\mathbb{C}^n$ as an adjoint orbit $\mathcal{O}_\lambda$ inside the Lie algebra of $n \times n$ skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of $\mathcal{O}_\lambda$, and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type $A$) a map of Bloch, Flaschka, and Ratiu (1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on $\mathcal{O}_\lambda$ which preserve positivity, working in three natural Riemannian metrics. In the K\"ahler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (2017, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2017, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on $\mathcal{O}_\lambda$. We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the K\"ahler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).