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Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties

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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 (Duke Math. J. 61(1): 41–65, 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 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 \({\mathcal {O}}_{\lambda }\). 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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Acknowledgements

We thank Roger Brockett, Sergey Fomin, Pavel Galashin, Thomas Lam, and Tudor Raţiu for valuable discussions, and Jonathan Boretsky for helpful feedback on the paper.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, USA

    Anthony M. Bloch

  2. Department of Mathematics, University of Notre Dame, Notre Dame, USA

    Steven N. Karp

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Correspondence to Steven N. Karp.

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A.M.B. was partially supported by NSF grants DMS-1613819 and DMS-2103026, and AFOSR grant FA 9550-22-1-0215. S.N.K. was partially supported by an NSERC postdoctoral fellowship.

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Bloch, A.M., Karp, S.N. Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties. Commun. Math. Phys. 398, 1213–1289 (2023). https://doi.org/10.1007/s00220-022-04540-5

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