skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • NSF Public Access
  • Search Results
  • Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties Anthony M. Bloch & Steven N. Karp
Title: Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties Anthony M. Bloch & Steven N. Karp
One can view a partial flag variety in ℂ𝑛 as an adjoint orbit 𝜆 inside the Lie algebra of 𝑛×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 𝜆 , 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 (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on 𝜆 which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 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 (2022, 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 𝜆 . We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).  more » « less
Award ID(s):
2103026
PAR ID:
10412942
Author(s) / Creator(s):
Date Published:
Journal Name:
Communications in Mathematical Physics
Volume:
398
ISSN:
0010-3616
Page Range / eLocation ID:
1213-1289
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Journal of the American Mathematical Society)
    We show that the totally nonnegative part of a partial flag variety G / P G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. 
    more » « less
  2. (, International Mathematics Research Notices)
    Abstract We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_f$$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of $$\operatorname{Crit}^{\geqslant 0}_f$$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_{k,n}$$ is homeomorphic to the second hypersimplex $$\Delta _{2,n}$$. 
    more » « less
  3. ; (, International journal of mathematics)
    As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non- perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. 
    more » « less
  4. ; ; (, Communications in Mathematical Physics)
    Abstract We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply thenonnegative configuration space. This space has a natural stratification bypositive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel. 
    more » « less
  5. ; (, Physcia D)
    The Toda lattice (1967) is a Hamiltonian system given by n points on a line governed by an exponential potential. Flaschka (1974) showed that the Toda lattice is integrable by interpreting it as a flow on the space of symmetric tridiagonal n × n matrices, while Moser (1975) showed that it is a gradient flow on a projective space. The symmetric Toda flow of Deift, Li, Nanda, and Tomei (1986) generalizes the Toda lattice flow from tridiagonal to all symmetric matrices. They showed the flow is integrable, in the classical sense of having d integrals in involution on its 2d-dimensional phase space. The system may be viewed as integrable in other ways as well. Firstly, Symes (1980, 1982) solved it explicitly via QR-factorization and conjugation. Secondly, Deift, Li, Nanda, and Tomei (1986) ‘tridiagonalized’ the system into a family of tridiagonal Toda lattices which are solvable and integrable. In this paper we derive their tridiagonalization procedure in a natural way using the fact that the symmetric Toda flow is diffeomorphic to a twisted gradient flow on a flag variety, which may then be decomposed into flows on a product of Grassmannians. These flows may in turn be embedded into projective spaces via Plücker embeddings, and mapped back to tridiagonal Toda lattice flows using Moser’s construction. In addition, we study the tridiagonalized flows projected onto a product of permutohedra, using the twisted moment map of Bloch, Flaschka, and Ratiu (1990). These ideas are facilitated in a natural way by the theory of total positivity, building on our previous work (2023). 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email