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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).
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This content will become publicly available on July 1, 2026
Higher Lorentzian Polynomials, Higher Hessians, and the Hodge–Riemann Relations for Graded Oriented Artinian Gorenstein Algebras in Codimension Two
Abstract We define $$i$$-Lorentzian polynomials in two variables, and characterize them as the real homogeneous Macaulay dual generators whose corresponding codimension two algebras satisfy the mixed Hodge–Riemann relations in degree $$i$$ on the positive orthant of linear forms. We further show that $$i$$-Lorentzian polynomials of degree $$d$$ are in one-to-one correspondence with totally nonnegative Toeplitz matrices of size $$(i+1)\times (d-i+1)$$. Using this latter characterization, we show that a certain subclass of real rooted polynomials called normally stable are $$i$$-Lorentzian for all $$i$$. Our results also lead to a new theorem on Toeplitz matrices: the closure of the set of totally positive Toeplitz matrices equals the set of totally nonnegative Toeplitz matrices.
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- PAR ID:
- 10611374
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 13
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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